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MATHEMATICAL MONOGRAPHS 

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PUBLISHED BY 
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CHAPMAN & HALL, Limited, LONDON 



MATHEMATICAL MONOGRAPHS. 

EDITED BY 

MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. 



No. 9. 



DIFFERENTIAL EQUATIONS. 



BY 

W. WOOLSEY JOHNSON, 

PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVAL ACADEMY. 



FOURTH EDITION. 
FIRST THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

LONDON: CHAPMAN & HALL, LIMITED. 

1906. 




COPYRIGHT, 1896, 

BY 
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD 

UNDER THE TlTLE 

HIGHER MATHEMATICS. 

First Edition, September, 1896. 
Second Edition, January, 1898. 
Third Edition, August, 1900. 
Fourth Edition, January, 1906. 



FOHRT DRUMMOND, PRINTRR, NRW YORIC. 



EDITORS' PREFACE. 



THE volume called Higher Mathematics, the first edition 
of which was published in 1896, contained eleven chapters by 
eleven authors, each chapter being independent of the others, 
but all supposing the reader to have at least a mathematical 
training equivalent to that given in classical and engineering 
colleges. The publication of that volume is now discontinued 
and the chapters are issued in separate form. In these reissues 
it will generally be found that the monographs are enlarged 
by additional articles or appendices which either amplify the 
former presentation or record recent advances. This plan of 
publication has been arranged in order to meet the demand of 
teachers and the convenience of classes, but it is also thought 
that it may prove advantageous to readers in special lines of 
mathematical literature. 

It is the intention of the publishers and editors to add other 
monographs to the series from time to time, if the call for the 
same seems to warrant it. Among the topics which are under 
consideration are those of elliptic functions, the theory of num- 
bers, the group theory, the calculus of variations, and non- 
Euclidean geometry; possibly also monographs on branches of 
astronomy, mechanics, and mathematical physics may be included.. 
It is the hope of the editors that this form of publication may 
tend to promote mathematical study and research over a wider 
field than that which the former volume has occupied. 

December, 1905. 



AUTHOR'S PREFACE. 



IT is customary to divide the Infinitesimal Calculus, or Calcu- 
lus of Continuous Functions, into three parts, under the heads 
Differential Calculus, Integral Calculus, and Differential Equa- 
tions. The first corresponds, in the language of Newton, to the 
"direct method of tangents, " the other two to the " inverse method 
of tangents"; while the questions which come under this last 
head he further -divided into those involving the two fluxions and 
one fluent, and those involving the fluxions and both fluents. 

On account of the inverse character which thus attaches to 
the present subject, the differential equation must necessarily 
at first be viewed in connection with a "primitive," from which 
it might have been obtained by the direct process, and the solu- 
tion consists in the discovery, by tentative and more or less arti- 
ficial methods, of such a primitive, when it exists; that is to 
say, when it is expressible in the elementary functions which 
constitute the original field with which the Differential Calculus 
has to do. 

It is the nature of an inverse process to enlarge the field of 
its operations, and the present is no exception; but the adequate 
handling of the new functions with which the field is thus enlarged 
requires the introduction of the complex variable, and is beyond 
the scope of a work of this size. 

But the theory of the nature and meaning of a differential 
equation between real variables possesses a great deal of interest. 
To this part of the subject I have endeavored to give a full treat- 
ment by means of extensive use of graphic representations in 



AUTHOR S PREFACE. V 

rectangular coordinates. If we ask what it is that satisfies an 
ordinary differential equation of the first order, the answer must 
be certain sets of simultaneous values of x, y, and p. The geo- 
metrical representation of such a set is a point in a plane asso- 
ciated with a direction, so to speak, an infinitesimal stroke, and 
the "solution" consists of the grouping together of these strokes 
into curves of which they form elements. The treatment of 
singular solutions, following Cayley, and a comparison with the 
methods previously in use, illustrates the great utility of this point 
of view. 

Again, in partial differential equations, the set of simultaneous 
values of x, y, z, p, and q which satisfies an equation of the first 
order is represented by a point in space associated with the direc- 
tion of a plane, so to speak by a flake, and the mode in which 
these coalesce so as to form linear surface elements and con- 
tinuous surfaces throws light upon the nature of general and 
complete integrals and of the characteristics. 

The expeditious symbolic methods of integration applicable 
to some forms of linear equations, and the subject of development 
of integrals in convergent series, have been treated as fully as space 
would allow. 

Examples selected to illustrate the principles developed in 
each section will be found at its close, and a full index of subjects 
at the end of the volume. 

W. W. J. 

ANNAPOLIS, MD., December, 1905. 



CONTENTS. 



ART. i. EQUATIONS OF FIRST ORDER AND DEGREE Page i 

2. GEOMETRICAL REPRESENTATION 3 

3. PRIMITIVE OF A DIFFERENTIAL EQUATION 5 

4. EXACT DIFFERENTLY EQUATIONS 6 

5. HOMOGENEOUS EQUATIONS 9 

6. THE LINEAR EQUATION 10 

7. FIRST ORDER AND SECOND DEGREE 12 

8. SINGULAR SOLUTIONS 15 

9. SINGULAR SOLUTION FROM THE COMPLETE INTEGRAL . . . 18 

10. SOLUTION BY DIFFERENTIATION 20 

11. GEOMETRIC APPLICATIONS; TRAJECTORIES 23 

12. SIMULTANEOUS DIFFERENTIAL EQUATIONS 25 

13. EQUATIONS OF THE SECOND ORDER 28 

14. THE Two FIRST INTEGRALS 31 

15. LINEAR EQUATIONS 34 

1 6. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 36 

17. HOMOGENEOUS LINEAR EQUATIONS 40 

18. SOLUTIONS IN INFINITE SERIES 42 

19. SYSTEMS OF DIFFERENTIAL EQUATIONS 47 

20. FIRST ORDER AND DEGREE WITH THREE VARIABLES ... .50 

21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE 53 

22. COMPLETE AND GENERAL INTEGRALS 57 

23. COMPLETE INTEGRAL FOR SPECIAL FORMS 60 

24. PARTIAL EQUATIONS OF SECOND ORDER 63 

25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS 66 

INDEX 72 



DIFFERENTIAL EQUATIONS. 



ART. 1. EQUATIONS -OF FIRST ORDER AND DEGREE. 

In the Integral Calculus, supposing y to denote an unknown- 
function of the independent variable x, the derivative of y with 
respect to x is given in the form of a function of x, and it is 
required to find the value of y as a function of x. In other 
words, given an equation of the form 

g=/O), or dy = flx)dx t (I) 

of which the general solution is written in the form 

y = //(*X*, (2) 

it is the object of the Integral Calculus to reduce the expres- 
sion in the second member of equation (2) to the form of a 
known function of x. When such reduction is not possible, 
the equation serves to define a new function of x. 

In the extension of the processes of integration of which 
the following pages give a sketch the given expression for the 
derivative may involve not only x, but the unknown function 
y ; or, to write the equation in a form analogous to equation 

(i), it may be 

Mdx + Ndy = o, (3) 

in which J/and TV are functions of x and y. This equation is 
in fact the general form of the differential equation of the first 
order and degree ; either variable being taken as the independ- 
ent variable, it gives the first derivative of the other variable 



2 DIFFERENTIAL EQUATIONS. 

in terms of x and y. So also the solution is not necessarily an 
expression of either variable as a function of the other, but is 
generally a relation between x and y which makes either an 
implicit function of the other. 

When we recognize the left member of equation (3) as an 
" exact differential," that is, the differential of some function of 
x and y, the solution is obvious. For example, given the equa- 
tion 

xdy-\-ydx = o, (4) 

the solution xy = C, (5) 

where C is an arbitrary constant, is obtained by " direct inte- 
gration." When a particular value is attributed to (7, the result 
is a " particular integral ; " thus^ = x~ l is a particular integral 
of equation (4), while the more general relation expressed by 
equation (5) is known as the " complete integral." 

In general, the given expression Mdx -j- Ndy is not an ex- 
act differential, and it is necessary to find some less direct 
method of solution. 

The most obvious method of solving a differential equation 
of the first order and degree is, when practicable, to " separate 
the variables," so that the coefficient of dx shall contain x 
only, and that of dy, y only. For example, given the equation 

( i y)dx -f- (i -{- x}dy = o, (6) 

the variables are separated by dividing by (\-\-x](\ y). 

dx dy 

Thus i h =0, 

I + x ' I y 

Each term is now directly integrable, and hence 
log (I +*) log (i y) = c. 

The solution here presents itself in a transcendental form, 
but it is readily reduced to an algebraic form. For, taking the 
exponential of each member, we find 

t ^ = c, whence i -J- x = C(i y) t (7) 

where C is put for the constant ^. 



GEOMETRICAL REPRESENTATION. 3 

To verify the result in this form we notice that differentia- 
tion gives dx = Cdy, and substituting in equation (6) we find 

- C(i -y) + i +* = o, 
which is true by equation (7). 

Prob. i. Solve the equation dy -\- y tan x dx = o. 

Ans. y=C cos x. 

Prob. 2. Solve ^ + '/ = a\ Ans. 



. 
dx by a 

-, , ^ / + 1 

Prob. 3. Solve -j- = =VT Ans. y = 



ex 



Prob. 4. Helrnholtz's equation for the strength of an electric 
current C at the time / is 

C = - - 

~ R R dt* 

where , R, and L are given constants. Find the value of C, de- 
termining the constant of integration by the condition that its initial 
value shall be zero. 



ART. 2. GEOMETRICAL REPRESENTATION. 

The meani'ng of a differential equation may be graphically 
illustrated by supposing simultaneous values of x and y to be 
the rectangular coordinates of a variable point. It is conven- 
ient to put/ for the value of the ratio dy-.dx. Then P being 
the moving point (x, y) and denoting the inclination of its 
path to the axis of x, we have 

dy 

p = -j- = tan 0. 
dx 

The given differential equation of the first order is a relation 
between/, x, andy, and, being of the first degree with respect 
to/, determines in general a single value of/ for any assumed 
values of x and y. Suppose in the first place that, in addition 
to the differential equation, we were given one pair of simul- 
taneous' values of x and y, that is, one position of the point P. 
Now let P start from this fixed initial point and begin to move 
in either direction along the straight line whose inclination 



DIFFERENTIAL EQUATIONS. 



is determined by the value of p corresponding to the initial 
values of x and y. We thus have a moving point satisfying 
the given differential equation. As the point P moves the 
values of x and y vary, and we must suppose the direction of 
its motion to vary in such a way that the simultaneous values 
of x, y, and/ continue to satisfy the differential equation. In 
that case, the path of the moving point is said to satisfy the 
differential equation. The point P may return to its initial 
position, thus describing a closed curve, or it may pass to infin- 
ity in each direction from the initial point describing an infinite 
branch of a curve.* The ordinary cartesian equation of the 
path of P is a particular integral of the differential equation. 

If no pair of associated values of x and y be known, P may 
be assumed to start from any initial point, so that there is an 
unlimited number of curves representing particular integrals 
of the equation. These form a "system of curves," and the 
complete integral is the equation of the system in the usual 
form of a relation between x, y, and an arbitrary " parameter." 
This parameter is of course the constant of integration. It is 
constant for any one curve of the system, and different values 
of it determine different members of the system of curves, or 
different particular integrals. 

As an illustration, let us take equation (4) of Art. I, which 

may be written 

d y___y_ 

dx x' 

Denoting by the inclination to 
the axis of x of the line joining P 
with the origin, the equation is 
equivalent to tan = tan 6, and. 
therefore expresses that P moves 
in a direction inclined equally with 
OP to either axis, but on the other 

* When the form of the functions M and N is unrestricted, there is no 
reason why either of these cases should exist, but they commonly occur among 
such differential equations as admit of solution. 




PRIMITIVE OF A DIFFERENTIAL EQUATION. 5 

side. Starting from any position in the plane, the point P 
thus moving must describe a branch of an hyperbola having 
the two axes as its asymptotes ; accordingly, the complete 
integral xy = C is the equation of the system consisting of 

these hyperbolas. 

ff^\ Lc- 

Prob. 5. Write the differential equation which requires P to move 
in a direction always perpendicular to OP, and thence derive the 
equation of the system of curves described. 

Ans ^-_*.*' + v'-C 
';<** ,'* + * 

* Prob. 6. What is the system described when is the comple- 
* . ment of ? 6 - Z^ Ans. x 9 -/ = C. 

oJ^jv^TProb. 7. If "^OsJ^, show geometrically that the system described 
5 ,^^consists of circles, and find the differential equation. 
^ 6 4(|F Ans. 2xydx = (x* y*)dy. 

ART. 3. PRIMITIVE OF A DIFFERENTIAL EQUATION. 

Let us now suppose an ordinary relation between x and y, 
which may be represented by a curve, to be given. By differ- 
entiation we may obtain an equation of which the given equa- 
, tion is of course a solution or particular integral. But by 
combining this with the given equation any number of differ- 
ential equations of which the given equation is a solution may 
be found. For example, from 

y = m(x a) (l) 

we obtain directly 

2ydy = mdx, (2) 

of which equation (i) is an integral; again, dividing (2) by (i) 
we have 

*** 




y x-a' 
and of this equation also (i) is an integral. 

If in equation (i) m be regarded as an arbitrary parameter, 
it is the equation of a system of parabolas having a common 
axis and vertex. The differential equation (3), which does not 
contain m, is satisfied by every member of this system of curves, 



DIFFERENTIAL EQUATIONS. 

Hence equation (i) thus regarded is the complete integral of 
equation (3), as will be found by solving the equation in which 
the variables are already separated. 

Now equation (3) is obviously the only differential equation 
independent of m which could be derived from (i) and (2), since 
it is the result of eliminating ;;/. It is therefore the " differ- 
ential equation of the system ; " and in this point of view the 
integral equation (i) is said to be its "primitive." 

Again, if in equation (i) a be regarded as the arbitrary con- 
stant, it is the equation of a system of equal parabolas having 
a common axis. Now equation (2) which does not contain a 
is satisfied by every member of this system of curves; hence it 
is the differential equation of the system, and its primitive is 
equation (i) with a regarded as the arbitrary constant. 

Thus, a primitive is an equation containing as well as x and" 
y an arbitrary constant, which we may denote by C, and the 
corresponding differential equation is a relation between x, y, 
and/, which is found by differentiation, and elimination of C if 
necessary. This is therefore also a method of verifying the com- 
plete integral of a given differential equation. For example, in 
verifying the complete integral (7) in Art. i we obtain by differ- 
entiation i = Cp. If we use this to eliminate C from equa- 
tion (7) the result is equation (6); whereas the process before 
employed was equivalent to eliminating / from equation (6), 
thereby reproducing equation (7). 

Prob. 8. Write the equation of the system of circles in Prob. 7, 
Art. 2, and derive the differential equation from it as a primitive. 

Prob. 9. Write the equation of the system of circles passing 
through the points (o, b} and (o, b], and derive from it the differ- 
ential equation of the system. 

ART. 4. EXACT DIFFERENTIAL EQUATIONS. 

In Art. I the case is mentioned in which Mdx -(- Ndy is an 
" exact differential," that is, the differential of a function of x 
andj. Let u denote this function; then 

(i) 



EXACT DIFFERENTIAL EQUATIONS. 

and in the notation of partial derivatives 

M=* t N=^. 
'dx -dy 

3* 
Then, since by a theorem of partial derivatives r- 



-dy -dx 

This condition rnjj^t therefore be fulfilled by M and N in 
order that equation (i) may be possible. When it is fulfilled 
Mdx -\- Ndy o is said to be an " exact differential equation," 
and its complete integral is 

u = C. (3) 

For example, given the equation 

x(x -\- 2y]dx -f- (x* y*}dy = o, 



= 2x, and - = 2*; the 



condition (2) is fulfilled, and the equation is exact. To find the 
function ?/, we may integrate Mdx, treating y as a constant; thus, 

&+Sy = Y, 

in which the constant of integration Y may be a function of y. 
The result of differentiating this is 

;* 

- x*dx -\- 2xy dx -f- x*dy = dY, 

which should be identical with the given equation ; therefore, 
dY = y*dy, whence Y = $y a -}- C, and substituting, the com- 
plete integral may be written 



The result is more readily obtained if we notice that all | 
terms containing x and dx only, or y and dy only, are exact w 
differentials; hence it is only necessary to examine the termsj 
containing both x and y. In the present case, these are 
2xy dx -\- x*dy, which obviously form the differential of x*y ; 
whence, integrating and multiplying by 3, we obtain the result 
above. 

The complete integral of any equation, in whatever way it 



'8 DIFFERENTIAL EQUATIONS. 

was found, can be put in the form u = C, by solving for C. 
Hence an exact differential equation du = o can be obtained, 
which must be equivalent to the given equation 

Mdx -f- Ndy o, (4) 

here supposed not to be exact. The exact equation du = o 
must therefore be of the form 

}ji(Mdx + Ndy) = o, (5) 

where /* is a factor containing at least one of the variables x 
and/. Such a factor is called an " integrating factor" of the 
given equation. For example,- the result of differentiating 
equation (7), Art. I, when put in the form u = C, is 

(i -y)dx + (\ 4- x)dy _ 

(i - yf 

so that (i y)~* is an integrating factor of equation (6). It 
is to be noticed that the factor by which we separated the 
variables, namely, (i y)~\i x)~ l , is also an integrating 
factor. 

It follows that if an integrating factor can be discovered, 
the given differential equation can at once be solved.* Such 
a factor is sometimes suggested by the form of the equation. 

Thus, given (y x)dy -\-ydx = o, 

the terms ydx xdy, which contain both x and y, are not ex- 
act, but become so when divided by either x 1 or y*\ and be- 
cause the remaining term contains^ only, j/~ 2 is an integrating 
factor of the whole expression. The resulting integral is 

ig.y + - = c. 

Prob. 10. Show from the integral equation in Prob. 9, Art. 3, that 
x~* is an integrating factor of the differential equation. 

Prob. n. Solve the equation x(x' 1 + 37*)^ -\-y(y* -f 3* a )^ = o. 

Ans. x* 



* Since f*M and uN in the exact equation (5) must satisfy the condition (2), 
we have a partial differential equation for //; but as a general method of finding 
ft this simply comes back to the solution of the original equation. 



HOMOGENEOUS EQUATION. 



, xdyydx 
Prob. 12. Solve the equation ydy -\-xdx -j -- . , . = o. 

* 
(Ans. 



2 X 

Prob. 13. If w = c is a form of the complete integral and p the 
corresponding integrating factor, show that l*f(u) is the general 
expression for the integrating factors. 

Prob. 14. Show that the expression x a }P(mydx + nxdy) has the 
integrating factor jp**- 1 -*^**- 1 .-^; an d by means of such a factor 
solve the equation y(y* + 2x*)dx -(- jc(^ 4 2y)</v = o. 

Ans. 2x*y y* = ex*. V 

Prob. 15. Solve (x 1 -{- y'}dx 2xydy o. Ans. x* y* = ex. 

ART. 5. HOMOGENEOUS EQUATION. 

The differential equation Mdx -f- Ndy = o is said to be 
homogeneous when M and N are homogeneous functions of 
v and y of the same degree ; or, what is the same thing, when 

dy y 

Hr is expressible as a function of . If in such an equation 

the variables are changed from x and y to x and v, where 

y 

v = whence y = xv and dy = xdv -f- vdx, 

the variables x and # will be separable. For example, the 

equation 

(x 2y)dx -f- ydy = o 

is homogeneous; making the substitutions indicated and 
dividing by x, 

(i 2v)dx -f- v(xdv -|- ^^f) = O, 

dx vdv 

whence -- \- -, -. = o. 

x ' (y i/ 

Integrating, log x + log (v - i) = C\ 

and restoring^, 

x 
log ( _y *) -- - = C. 

' y x 

The equation Mdx -f- -/VWy = o can always be solved when 



10 DIFFERENTIAL EQUATIONS. 

M and N are functions of the first degree, that is, when it is 
of the form 

(ax + fy + c]dx + (a'x -f b'y + c'}dy = o. 
For, assuming x = x' -f- //, y = y' -f- ^> it becomes 
(*'+ b'y'+ ah + bk+ c)dx'+(a'x'+ b'y'+ a'h +b'k+c')dy' =o, 
which, by properly determining h and k, becomes 

(ax' + /X*' + (a'x' -f- '/>//, 
a homogeneous equation. 

This method fails when a : b == a' : b', that is, when the 
aquation takes the form 

(ax -\-by-\- c)dx -\- \m(ax -f- by) -\- c'~\dy = o ; 

but in this case if we put z = ax -f- by, and eliminate y, it will 
be found that the variables x and 2 can be separated. 

Prob. 16. Show that a homogeneous differential equation repre- 
sents a system of similar and similarly situated curves, the origin 
being the center of similitude, and hence that the complete integral 
may be written in a form homogeneous in x, y, and c. 

Prob. 17. Solve xdy ydx y(x* -f y*)dx = o. 

Ans. x* = * 2cy. 

Prob. 1 8. Solve (3^ 7* + i)dx + (jy 3* + $)dy = o. 

Ans. (y x + r)'(7 + x i) 5 = r. 

Prob. 19. Solve (x*.-{-y*)dx zxydy = o. Ans. .r a y 9 = ex. * 

Prob. 20. Solve (i + xy)ydx + (i tfy).*^ = o by introducing 



_ 
the new variable z = xy. Ans. x = Cye*y. 

Prob. 21. Solve ~ ax -\-by-\-c. Ans. ata+^-i- -!-& = <?**. 

ART. 6. THE LINEAR EQUATION. 

A differential equation is said to be "linear" when (one of 
the variables, say x, being regarded as independent,) it is of 
the first degree with respect to y, and its derivatives. The 
linear equation of the first order may therefore be written in 
the form 



THE LINEAR EQUATION. 11 

where P and Q are functions of x only. Since the second 
member is a function of x, an integrating factor of the first 
member will be an integrating factor of the equation provided 
it contains x only. To find such a factor, we solve the equation 



dy 
which is done by separating the variables ; thus, = Pdx \ 

whence log y = c I Pdx or 

y = Ce ~' M "- (3) 

Putting this equation in the form u = c, the corresponding 
exact equation is 

e SPd \dy + Pydx} = o, 

whence tr ' ' is the integrating factor required. Using this 
factor, the general solution of equation (i) is 

Qdx + C. (4) 



In a given example the integrating factor should of course 
be simplified in form if possible. Thus 

(i -(- x*}dy = (m -\- xy)dx 
is a linear equation for/; reduced to the form (i), it is 

dy x m 

dx i -\-a? i -f- x" 
from which 



The integrating factor is, therefore, 



e fpdx = 



whence the exact equation is 

dy xy dx mdx 



r 
J 
I 



12 DIFFERENTIAL EQUATIONS. 

Integrating, there is found 

y w* ~ 

v(i+O =: i/(i + *') + C) 

or 

y = mx -\- C \/(l + X*). 

ri r An equation is sometimes obviously linear, not for^, but 

for some function of y.' For example, the equation 

dy . 

-j \- tan y = x sec y 

when multiplied by cos y takes a form linear for sin y ; the 
integrating factor is e*, and the complete integral 
sin y = x i -|- ce~ x . 

dy 
In particular, the equation ^- -j- Py = Qy* t which is known as 

" the extension of the linear equation," is readily put in a form 
linear for y l ~ M . 

d y , . 

rob. 22. Solve ar-r- + (i 2#)y = ^ a - Ans. y = x*(\ + ce*). 



dy 
Prob. 23. Solve cos x - -{-y i -(- sin ^r = o. 

Ans. Xsec * + tan *) = * + ^ 

A 

Prob. 24. Solve cos x + y sin ^ = i. 
rf^ 

Ans. y = sin * + c cos ^c. 
Prob. 25. Solve -^ = *V xy. Ans. 4 = ** + i +^. 



Prob. 26. Solve ^ = - ,' , 3 . Ans. - = 2 -/+ 
dx yy + 0:7 a: 



. 

y 



ART. 7. FIRST ORDER AND SECOND DEGREE. 

If the given differential equation of the first order, or re- 
lation between x, y, and /, is a quadratic for p, the first step 
in the solution is usually to solve for /. The resulting value 
of / will generally involve an irrational function of x and y\ 
>so that an equation expressing such a value of /, like some of 
those solved in the preceding pages, is not properly to be re- 



FIRST ORDER AND SECOND DEGREE. 13 

garded as an equation of the first degree. In the exceptional 
case when the expression whose root is to be extracted is a 
perfect square, the equation is decomposable into two equa- 
tions properly of the first degree. For example, the equation 



y x 

when solved for / gives 2p = -, or 2p = ; it may therefore 

be written in the form 

(2px - y)(2py - x) = Q, 
and is satisfied by putting either 

dy _ y dy _ x 

~T~ - OF ~7~ - - 

ax 2x ax 2y 

The integrals of these equations are 

y* = ex and 2y* x 1 = C, 

and these form two entirely distinct solutions of the given 
equation. 

As an illustration of the general case, let us take the equation 



Separating the variables and integrating, 

Vic Vy = Vc, (2) 

and this equation rationalized become^ 

(x- y y-2c(x+y)+c*=o. (3) 

There is thus a single complete integral containing one arbi- 
trary constant and representing a single system of curves; 
namely, in this case, a system of parabolas touching each axis 
at the same distance c from the origin. The separate equa- 
tions given in the form (2) are merely branches of the same 
parabola. 

Recurring now to the geometrical interpretation of a differ- 
ential equation, as given in Art. 2, it was stated that an equa- 
tion of the first degree determines, in general, for assumed 
values of x and y, that is, at a selected point in the plane, a 
single value of p. The equation was, of course, then supposed 



14 DIFFERENTIAL EQUATIONS. 

rational in x and y* The only exceptions occur at points for 
which the value of p takes the indeterminate form ; that is, 
the equation being Mdx -\- Ndy = o, at points (if any exist) 
for which M = o and N = o. It follows that, except at such 
points, no two curves of the system representing the complete 
integral intersect, while through such points an unlimited num- 
ber of the curves may pass, forming a "pencil of curves." f 

On the other hand, in the case of an equation of the second 
cdegree, there will in general be two values of / for any given 
point. Thus from equation (i) above we find for the point 
(4, i), p = ; there are therefore two directions in which a 
point starting from the position (4, i) may move while satis- 
fying the differential equation. The curves thus described 
represent two of the particular integrals. If the same values 
of x and y be substituted in the complete integral (3), the re- 
sult is a quadratic for c, giving c = 9 and c = i, and these 
determine the two particular integral curves, Vx -j- Vy = 3, 
and Vx Vy = i. 

In like manner the general equation of the second degree, 
which may be written in the form 



where L, M, and A 7 " are one-valued functions of x and y, repre- 
sents a system of curves of which two intersect in any given 
point for which p is found to have two real values. For these 
points, therefore, the complete integral should generally give 
two real values of c. Accordingly we may assume, as the 
standard form of its equation, 
PC* 



* In fact / was supposed to be a one-valued function of x and y; thus, 
/ = sin" 1 * would not in this connection be regarded as an equation of the first 
degree. 

f In Prob. 6, Art. 3, the integral equation represents the pencil of circles pass- 
ing through the points (o, b) and (o, b)\ accordingly/ in the differential equa- 
tion is indeterminate at these points. In some cases, however, such a point is 
merely a node of one particular integral. Thus in the illustration given in Art. 2, 
/ is indeterminate at the origin, and this point is a node of the only particular 
integral, xy = o, which passes through it. 



SINGULAR SOLUTIONS. 15 

where P, Q, and R are also one-valued functions of x and y. 
If there are points which make / imaginary in the differential 
equation, they will also make c imaginary in the integral. 

Prob. 27. Solve the equation /" +y = i and reduce the inte- ./ 
gral to the standard form. 

Ans. (y -f- cos x)^ 2C sin x -f- y cos x = o. 

Prob. 28. Solve yp* + zxp y = o, and show that the intersect- 
ing curves at any given point cut at right angles. , 

Prob. 29. Solve (x 9 + i)/ = i. Ans. <rV r - text? = i. 

ART. 8. SINGULAR SOLUTIONS. 

A differential equation not of the first degree sometimes 
admits of what is called a " singular solution ; " that is to say, a 
solution which is not included in the complete integral. For 
suppose that the system of curves representing the complete 
integral has an envelope. Every point A of this envelope 
is a point of contact with a particular curve of the complete in- 
tegral system ; therefore a point moving in the envelope when 
passing through A has the same values of x, y, and /as if it 
were moving through A in the particular integral curve. Hence 
such a point satisfies the differential equation and will continue 
to satisfy it as long as it moves in the envelope. The equation 
of the envelope is therefore a solution of the equation. 

As an illustration, let us take the system of straight lines 
whose equation is 

i a 

y = cx + -, (I) 

C 

where c is the arbitrary parameter. The differential equation 
derived from this primitive is 

y=px + - p * (2) 

of which therefore (i) is the complete integral. 

Now the lines represented by equation (i), for different 
values of c, are the tangents to the parabola 

/ = V**- (3) 



1G 



DIFFERENTIAL EQUATIONS. 



A point moving in this parabola has the same value of/ as if it 

were moving in one of the tan- 
gents, and accordingly equation 
(3) will be found to satisfy the 
differential equation (2). 

It will be noticed that for 
any point on the convex side of 
the parabola there are two real 
values of p ; for a point on the 
other side the values of / are 
imaginary, and for a point on 
the curve they are equal. Thus 
its equation (3) expresses the 
relation between x and y which must exist in order that (2) 
regarded as a quadratic for p may have'equal roots, as will be 
seen on solving that equation. 

In general, writing the differential equation in the form 




the condition of equal roots is 



o. (5) 

The first member of this equation, which is the " discrimi- 

nant " of equation (4), frequently admits of separation into 

factors rational in x and_y. Hence, if there be a singular solu- 

tion, its equation will be found by putting the discriminant of 

the differential equation, or one of its factors, equal to zero. 

It does not follow that every such equation represents a solu- 
tion of the differential equation. It can only be inferred that 
it is a locus of points for which the two values of / become 
equal. Now suppose that two distinct particular integral 
curves touch each other. At the point "of contact, the two 
values of/, usually distinct, become equal. The locus of such 
points is called a "tac-locus." Its equation plainly satisfies the 
discriminant, but does not satisfy the differential equation. An 
illustration is afforded by the equation 



SINGULAR SOLUTIONS. 17 

of which the complete integral isy 1 -f- (x c)* = a*, and the 
discriminant, see equation (5), is j"(y tf 2 ) = o. 

This is satisfied by y = a, y a, and y = o, the first two 
of which satisfy the differential equation, while U = o does not. 
The complete integral represents in this case all circles of radius 
a with center on the axis of x. Two of these circles touch at 
every point of the axis of x, which is thus a tac-locus, while 
y = a and y = a constitute the envelope. 

The discriminant is the quantity which appears under the 
radical sign when the general equation (4) is solved for/, and 
therefore it changes sign as we cross the envelope. But the 
values of / remain .real as we cross the tac-locus, so that the 
discriminant cannot change sign. Accordingly the factor which 
indicates a tac-locus appears with an even exponent (as y 1 in 
the example above), whereas the factor indicating the singular 
solution appears as a simple factor, or with an odd exponent. 

A simple factor of the discriminant, or one with an odd ex- 
ponent, gives in fact always the boundary between a region of 
the plane in which / is real and one in which p is imaginary ; 
nevertheless it may not give a singular solution. For the two 
arcs of particular integral curves which intersect in a point on 
the real side of the boundary may, as the point is brought up 
to the boundary, become tangent to each other, but not to the 
boundary curve. In that case, since they cannot cross the 
boundary, they become branches of the same particular inte- 
gral forming a cusp. A boundary curve of this character is 
called a "cusp-locus" ; the value of / for a point moving in it 
is of course different from the equal values of/ at the cusp, and 
therefore its equation does not satisfy the differential equation.* 

Prob. 30. To what curve is the line y = mx -\- a |/(i m*) 
always tangent ? Ans. y 1 x* = a*. 

Prob. 31. Show that the discriminant of a decomposable differ- 

* Since there is no reason why the values of/ referred to should be identical, 
we conclude that the equation Z/ 9 -f Mp + N = o has not in general a singular 
solution, its discriminant representing a.cusp-locus except when a certain con- 
dition is fulfilled. 



18 DIFFERENTIAL EQUATIONS. 

cntial equation cannot be negative. Interpret the result of equating 
it to zero in the illustrative example at the beginning of Art. 7. 

Prob. 32. Show that the singular solutions of a homogeneous dif- 
ferential equation represent straight lines passing through the origin. 

Prob. 33. Solve the equation xp* zyp -\- ax o. 

Ans. x* 2cy -f- af = o ; singular solution y = ax*. 

Prob. 34. Show that the equation /" -f- zxp y = o has no sin- 
gular solution, but has a cusp-locus, and that the tangent at every 
cusp passes through the origin. 

ART. 9. SINGULAR SOLUTION FROM COMPLETE INTEGRAL. 

When the complete integral of a differential equation of 
the second degree has been found in the standard form 

PS+Qc + R = o (i) 

(see the end of Art. 7), the substitution of special values of x 
and y in the functions P, Q, and R gives a quadratic for c whose 
roots determine the two particular curves of the system which 
pass through a given point. If there is a singular solution, 
that is, if the system of curves has an envelope, the two 
curves which usually intersect become identical when the given 
point is moved up to the envelope. Every point on the en- 
velope therefore satisfies the condition of equal roots for equa- 
tion (i), which is 

Q- 4/^ = 0; (2) 

and, reasoning exactly as in Art. 8, we infer that the equation 
of the singular solution will be found by equating to zero the 
discriminant of the equation in c or one of its factors. Thus 
the discriminant of equation (i), Art. 8, or "^-discriminant," is 
the same as the "/-discriminant," namely, y ^ax, which 
equated to zero is the equation of the envelope of the system of 
straight lines. 

But, as in the case of the /-discriminant, it must not be 
inferred that every factor gives a singular solution. For ex- 
ample, suppose a squared factor appears in the ^-discriminant. 
The locus on which this factor vanishes is not a curve in cross- 
ing which c and/ become imaginary. At any point of it there 



SINGULAR SOLUTION FROM COMPLETE INTEGRAL. 19 

\vill be two distinct values of p, corresponding to arcs of par- 
ticular integral curves passing through that point ; but, since 
there is but one value of c, these arcs belong to the same par- 
ticular integral, hence the point is a double point or node. 
The locus is therefore called a " node-locus." The factor repre- 
senting it does not appear in the /-discriminant, just as that 
representing a tac-locus does not appear in the <r-discriminant. 

Again, at any point of a cusp-locus, as shown at the end of 
Art. 8, the two branches of particular integrals become arcs ot 
the same particular integral ; the values of c become equal, so 
that a cusp-locus also makes the ^-discriminant vanish. 

The conclusions established above obviously apply also to 
equations of a degree higher than the second. In the case of 
the <r-equation the general method of obtaining the condition 
for equal roots, which is to eliminate c between the original and 
the derived equation, is the same as the process of finding the 
envelope or "locus of the ultimate intersections" of a, system 
of curves in which c is the arbitrary parameter. 

Now suppose the system of curves to have for all values of 
* a double point, it is obvious that among the intersections 
of two neighboring curves there are two in the neighborhood 
of the nodes, and that ultimately they coincide with the node, 
which accounts for the node-locus appearing twice in the dis- 
criminant or locus of ultimate intersections. In like manner, 

* It is noticed in the second foot-note to Art. 7 that for an equation of the 
first degree p takes the indeterminate form, not only at a point through which all 
curves of the system pass (where the value of c would also be found indeter- 
minate), but at a node of a particular integral. So also when the equation is of 
the th degree, if there is a node for a particular value of c, the n values of e at 
the point (which is not on a node-locus where two values of c are equal) deter- 
mine n -f- i arcs of particular integrals passing through the point ; and there- 
fore there are n -\- i distinct values of / at the point, which can only happen 
when p takes the indeterminate form, that is to say, when all the coefficients of 
the/-eguation (which is of the wth decree} vanish. See Cayley on Singular So- 
lutions' in the Messenger of Mathematics. New Series, Vol. II, p. 10 (Collected 
Mathematical Works, Vol. VIII, p. 529). The present t-neory of Singular Solu- 
tions was established by Cayley in this paper and its continuation, Vol. VI, p. 23. 
See also a paper by Dr. Glaisher, Vol. XII, p. I. 



20 DIFFERENTIAL EQUATIONS. 

if there is a cusp for all values of c, there are three intersections 
of neighboring curves (all of which may be real) which ulti- 
mately coincide with the cusp ; therefore a cusp-locus will 
appear as a cubed factor in the discriminant.* 

Prob. 35. Show that the singular solutions of a homogeneous 
equation must be straight lines passing through the origin. 

Prob. 36. Solve 3/V 2xyp + 47* x* = o, and show that there 
is a singular solution and a tac-focus. 

Prob. 37. Solve yp*-\- 2Xp y = o t and show that there is an 
imaginary singular solution. Ans. y* = zcx + c 1 . 

Prob. 38. Show that the equation (i x^p' = i y* represents 
a system of conies touching the four sides of a square. 

Prob. 39. Solve yp* ^xp -\-y = o ; examine and interpret both 
discriminants. Ans. c 1 + 2cx($y' 8.x 2 ) 3* a y +/* = o. 



ART. 10. SOLUTION BY DIFFERENTIATION. 

The result of differentiating a given differential equation o 
the first order is an equation of the second order, that is, it 

d*y 
contains the derivative -r-^ ; but, if it does not contain y ex- 

plicitly, it may be regarded as an equation of the first order for 
the variables x and/. If the integral of such an equation can 
be obtained it will be a relation between x, p, and a constant 
of integration c, by means of which / can be eliminated from 
the original equation, thus giving the relation between x, y> 
and c which constitutes the complete integral. For example, 
the equation 

(i) 



* The discriminant of PC* -\- Qc + R = o represents in general an envelope, 
no further condition requiring to be fulfilled as in the case of the discriminant 
of Z/ 2 -f- Mp -f- -A/ = o. Compare the foot-note to Art. 8. Therefore where 
there is an integral of this form there is generally a singular solution, although 
Z/ 9 -f- Mp -}- .A 7 = o has not in general a singular solution. We conclude, there- 
fore, that this equation (in which Z, M, and N are one-valued functions of x 
and y) has not in general an integral of the above form in which P, Q, and R 
are one-valued functions of x and y. Cayley, Messenger of Mathematics, New 
Series, Vol. VI, p. 23. 



SOLUTION BY DIFFERENTIATION. 21 

when solved for^, becomes 

y = x + ^p\ (2) 

whence by differentiation 

The variables can be separated in this equation, and its inte- 
gral is 



Substituting in equation (2), we find 



which is the complete integral of equation (i). 

This method sometimes succeeds with equations of a higher 
degree when the solution with respect to p is impossible or 
leads to a form which cannot be integrated. A differential 
equation between p and one of the two variables will be ob- 
tained by direct integration when only one of the variables is 
explicitly present in the equation, and also when the equation 
is of the first degree with respect to x and y. In the latter 
case after dividing by the coefficient of y, the result of differ- 
entiation will be a linear equation for x as a function of p, so 
that an expression for x in terms of p can be found, and then 
by substitution in the given equation an expression for y in 
terms of p. Hence, in this case, any number of simultaneous 
values of x and y can be found, although the elimination of p 
may be impracticable. 

In particular, a homogeneous equation which cannot be 
solved for p may be soluble for the ratio y : x, so as to assume 
the form y = x(f>(p). The result of differentiation is 



in which the variables x and p can be separated. 
Another special case is of the form 

y = P* +/<, (0 



22 DIFFERENTIAL EQUATIONS. 

which is known as Clairaut's equation. The result of differ- 
entiation is 



which implies either 

=o, or 



The elimination of p from equation (i) by means of the 
first of these equations * gives a solution containing no arbi- 
trary constant, that is, a singular solution. The second is a 
differential equation for/; its integral is p = c, which in 
equation (i) gives the complete integral 

y = c X +f(c\ (2} 

This complete integral represents a system of straight lines, 
the singular solution representing the curve to which they are 
all tangent. An example has already been given in Art. 8. 

A differential equation is sometimes reducible to Clairaut's 
form by means of a more or less obvious transformation of the 
variables. It may be noticed in particular that an equation of 

the form 

y = nx p _}_ (j)( x , p) 

is sometimes so reducible by transformation to the independent 
variable z, where x = z n ; and an equation of the form 



by transformation to the new dependent variable v y n . A 
double transformation of the form indicated may succeed 
when the last term is a function of both x and y as well as of/. 

Prob. 40. Solve the equation $y = 2/ 3 + 3/ 2 ; find a singular 
solution and a cusp-locus. Ans. (x -j- y -\- c i) a == ( xJ t~ f Y' 

a 
Prob. 41. Solve zy = xp -\ , and find a cusp-locus. 

Ans. aV \2acxy -f- Scy 3 i2x*y* -\- i6ax* == o. 

* The equation is in fact the same that arises in the general method for the 
condition of equal roots. See Art. 9. 



GEOMETRIC APPLICATIONS ; TRAJECTORIES. 23 

Prob. 42. Solve (x* (?}p* 2xyp + / a* = o. 

Ans. The circle x* + / = a\ and its tangents. 

Prob. 43. Solve y = xp + *y. 

Ans. ^ -j- <r .ay = o, and i + $x*y = o. 



Prob. 44. Solve / s $xyp + By 1 = o. 

Ans. y = c(x c}* 2jy = 43? and y = o are singulat 
solutions \y = o is also a particular integral. 

Prob. 45. Solve x\y px] = yp\ Ans. / = ex* + c\ 

ART. 11. GEOMETRIC APPLICATIONS ; TRAJECTORIES. 

Every property of a curve which involves the direction of 
its tangents admits of statement in the form of a differential 
equation. The solution of such an equation therefore deter- 
mines the curve having the given property. Thus, let it be 
required to determine the curve in which the angle between 
the radius vector and the tangent is n times* the vectorial 
angle. Using the expression for the trigonometric tangent of 
that angle, the expression of the property in polar coordi- 
nates is 

rdB 

= tan nO. 
dr 

Separating the variables and integrating, the complete 

integral is 

r" = C H sin nB. 

The mode in which the constant of integration enters here 
shows that the property in question is shared by all the mem- 
bers of a system of similar curves. 

The solution of a question of this nature will thus in gen- 
eral be a system of curves, the complete integral of a differential 
equation, but it may be a singular solution. Thus, if we ex- 
press the property that the sum of the intercepts on the axes 
made by the tangent to a curve is equal to the constant a, the 
straight lines making such intercepts will themselves consti- 
tute the complete integral system, and the curve required is 
the singular solution, which, in accordance with Art. 8, is the 



24 DIFFERENTIAL EQUATIONS. 

envelope of these lines. The result in this case will be found 
to be the parabola Vx ~\- \^y = Va. 

An important application is the determination of the 
"orthogonal trajectories" of a given system of curves, that is 
to say, the curves which cut at right angles every\curve of the 
given system. The differential equation of the trajectory is 
readily derived from that of the given system ; for at every 
point of the trajectory the value of p is the negative reciprocal 
of its value in the given differential equation. We have there- 
fore only to substitute /"' for p to obtain the differential 
equation of the trajectory. For example, let it be required to 
determine the orthogonal trajectories of the system of pa- 
rabolas 



having a common axis and vertex. The differential equation 
of the system found by eliminating a is 

2 xdy = y dx. 

Putting -- in place of -7-, the differential equation oi 
dy dx 

the system of trajectories is 

2.x dx -\- ydy = o, 
Whence, integrating, 



, The trajectories are therefore a system of similar ellipses 
with axes coinciding with the coordinate axes. 

Prob. 46. Show that when the differential equation of a system 
is of the second degree, its discriminant and that of its trajectory 
system will be identical ; but if it represents a singular solution in 
one system, it will constitute a cusp locus of the other. 

Prob. 47. Determine the curve whose subtangent is constant and 
equal to a. Ans. ce*=y*. 

Prob. 48. Show that the orthogonal trajectories of the curves 

r n =c" sin## are the same system turned through the angle about 

2n 

the pole. Examine the cases n = i, n = 2, and n = . 

Prob. 49. Show that the orthogonal trajectories of a system of 



SIMULTANEOUS DIFFERENTIAL EQUATIONS. 25 

circles passing through two given points is another system of circles 
having a common radical axis. 

Prob. 50. Determine the curve such that the area inclosed 
by any two ordinates, the curve and the axis of x, is equal to 
the product of the arc and the constant line a. Interpret the 

singular solution. 

Ans. The catenary ^ = \a(e a e a ). 

Prob. 51. Show that a system of confocal conies is self-orthog- 
onal. 

ART. 12. SIMULTANEOUS DIFFERENTIAL EQUATIONS. 

A system of equations between n -j- I variables and their 
differentials is a " determinate" differential system, because it 
serves to determine the n ratios of the differentials ; so that, 
taking any one of the variables as independent, the others vary 
in a determinate manner, and may be regarded as functions of 
the single independent variable. Denoting the variables by .#, 
y, 2, etc., the system may be written in the symmetrical form 

dx _ dy _ dz _ 

~X~~^Y~~Z~- "' 

where X, Y, Z . . . may be any functions of the variables. 

If any one of the several equations involving two differen- 
tials contains only the two corresponding variables, it is an 
ordinary differential equation ; and its integral, giving a re- 
lation between these two variables, may enable us by elimina- 
tion to obtain another equation containing two variables only, 
and so on until n integral equations have been obtained. 
Given, for example, the system 

dx_d^ _dz^ ,. 

x~ z " y' 

The relation between dy and dz above contains the varia- 
bles y and z only, and its integral is 

7' s? = a. (2) 

Employing this to eliminate z from the relation between 
dx and dy it becomes 

dx _ dy 

~' 



20 DIFFERENTIAL EQUATIONS. 

of which the integral is 



*) = *>*' (3) 

The integral equations (2) and (3), involving two constants 
of integration, constitute the complete solution. It is in like 
manner obvious that the complete solution of a system of n 
equations should contain n arbitrary constants. 

Confining ourselves now to the case of three variables, an 
extension of the geometrical interpretation given in Art. 2 
presents itself. Let x, y, and z be rectangular coordinates of 
P referred to three planes. Then, if P starts from any given 
position A, the given system of equations, determining the 
ratios dx : dy : dz, determines the direction in space in which P 
moves. As P moves, the ratios of the differentials (as deter- 
mined by the given equations) will vary, and if we suppose P 
to move in such a way as to continue to satisfy the differential 
equations, it will describe in general a curve of double curva- 
ture which will represent a particular solution. The complete 
solution is represented by the system of lines which may be 
thus obtained by varying the position of the initial point A. 
This system is a " doubly infinite " one ; for the two relations 
between x, y, and z which define it analytically must contain 
two arbitrary parameters, by properly determining which we 
can make the line pass through any assumed initial point.* 

Each of the relations between x, y and z, or integral equa- 
tions, represents by itself a surface, the intersection of the two 
surfaces being a particular line of the doubly infinite system. 
An equation like (2) in the example above, which contains only 
one of the constants of integration, is called an integral of the 
differential system, in contradistinction to an " integral equa- 

* It is assumed in the explanation that X, V, and Zare one-valued functions 
of x, y, and 2. There is then but one direction in which P can move when 
passing a given point, and the system is a non-intersecting system of lines. But 
if this is not the case, as for example when one of the equations giving the ratio 
of the differentials is of higher degree the lines may form an intersecting sys- 
tem, and there would be a theory of singular solutions, into which we do not 
here enter. 



SIMULTANEOUS DIFFERENTIAL EQUATIONS. 27 

tion " like (3), which contains both constants. An integral 
represents a surface which contains a singly infinite system of 
lines representing particular solutions selected from the doubly 
infinite system. Thus equation (2} above gives a surface on 
which lie all those lines for which a has a given value, while b 
may have any value whatever ; in other words, a surface which 
passes through an infinite number of the particular solution 
lines. 

The integral of the system which corresponds to the con- 
stant b might be found by eliminating a between equations (2) 
and (3). It might also be derived directly from equation (i) ; 
thus we may write 

dx^ _ dy _dz _ dy -J- dz _du 
x z y y ' -f- z u' 

in which a new variable u =. y -f- z is introduced. The rela- 
tion between dx and du now contains but two variables, and 

its integral, 

y + z = bx, (4) 

is the required integral of the system ; and this, together with 
the integral (2\ presents the solution of equations (i) in its 
standard form. The form of the two integrals shows that in 
this case the doubly infinite system of lines consists of hyper- 
bolas, namely, the sections of the system of hyperbolic cylinders 
represented by (2) made by the system of planes represented 

by (4). 

A system of equations of which the members possess a cer- 
tain symmetry may sometimes be solved in the following 
manner. Since 

dx _ dy _ dz _ \dx -f- pdy -f- vdz 
' 



if we take multipliers A, /v, v such that 



we shall have \dx -f- pdy -f- y dz = o. 

If the expression in the first member is an exact differential, 



28 DIFFERENTIAL EQUATIONS. 

direct integration gives an integral of the given system. For 
example, let the given equations be 

dx dy dz 

mz ny nx Iz ly mx ' 

/, m and n form such a set of multipliers, and so also do x, y 
and z. Hence we have 

Idx -j- mdy -f- ndz = o, 
and also xdx -\-ydy-\- zdz = o. 

Each of these is ai\ exact equation, and their integrals 

Ix -J- my -\- nz = a 
a nd x* -J- y* -j- z* = b* 

constitute the complete solution. The doubly infinite system 
of lines consists in this case of circles which have a common 
axis, namely, the line passing through the origin and whose 
direction cosines are proportional to /, m, and n. 

dx dy dz 
Prob. 132. Solve the equations -5 5 . = ^ = , and 

x y z 2xy 2xz 

interpret the result geometrically. (Ans. y=az, x*-\-y* -^-z^bz.} 

dx dy dz 

^Prob. 53. Solve 




x z 



Prob. 54. Solve ,, ~ = , ^ = , ^- . 

(b c}yz (c a)zx (a b)xy 

Ans. x* +/ + 2 a = A, ax' -f b? + c# - B. 

ART. 13. EQUATIONS OF THE SECOND ORDER. 

A relation between two variables and the successive deriva- 
tives of one of them with respect to the other as independent 
variable is called a differential equation of the order indicated 
by the highest derivative that occurs. For example, 



is an equation of the second order, in which x is the independent 



EQUATIONS OF THE SECOND ORDER. 29 

variable. Denoting as heretofore the first derivative by/, this 
equation may be written 

/ , v\^P_\ j. i _ / \ 

and this, in connection with 



which defines /, forms a pair of equations of the first order, 
connecting the variables x, y, and /. Thus any equation of the 
second order is equivalent to a pair of simultaneous equations 
of the first order. 

When, as in this example, the given equation "does not con- 
tain^ explicitly, the first of the pair of equations involves only 
the two variables x and/ ; and it is further to be noticed that, 
when the derivatives occur only in the first degree, it is a linear 
equation for/. Integrating equation (i) as such, we find 



and then using this value of/ in equation (2), its integral is 

y = c, - mx + c t log \x + y(i + *')], (4) 

in which, as in every case of two simultaneous equations of the 
first order, we have introduced two constants of integration. 

An equation of the first order is readily obtained also 
when the independent variable is not explicitly contained in 
the equation. The general equation of rectilinear motion in 

ffs 
dynamics affords an illustration. This equation is = f(s), 

where s denotes the distance measured from a fixed center of 

d*v 
force upon the line of motion. It may be written = f(s), in 

U-f 

connection with = v, which defines the velocity. Eliminat- 
dt 

ing dt from these equations, we have vdv = f(s)ds, whose 
integral is $v* = I f(s)ds -\- c, the "equation of energy" for 
the unit mass. The substitution of the value found for v in the 



30 DIFFERENTIAL EQUATIONS. 

second equation gives an equation from which t is found in 
terms of s by direct integration. 

The result of the first integration, such as equation (3) above, 

is called a "first integral" of the given equation of the second 

order ; it contains one constant of integration, and its complete 

integral, which contains a second constant, is also the "com- 

plete integral " of the given equation. 

A differential equation of the second order is " exact " when, 
all its terms being transposed to the first member, that member 
is the derivative with respect to x of an expression of the first 
order, that is, a function of x, y and p. It is obvious that the 
terms containing the second derivative, in such an exact differ- 
ential, arise solely from the differentiation of the terms con- 
taining/ in the function of x, y and/. For example, let it be 
required to ascertain whether 



is an exact equation. The terms in question are (i x*}-f-, 

ax 

which can arise only -from the differentiation of (i x^p. 
Now subtract from the given expression the complete deriva- 

tive of (i x*}p, which is 

, cTy dy 

/T _ y- \ - 2 X * 

\ / J a _/ 

ax ax 

the remainder is x -\- y, which is an exact derivative, namely, 
ax 

that of xy. Hence the given expression is an exact differ- 
ential, and 

(i-^ + ^y^ (6) 

is the first integral of the given equation. Solving thi-s linear 
-equation for y, we find the complete integral 

y = Cl x + c t tf(i *?). (7) 



,. 
\Prob. 55- Solve (i - * 2 ) 

Ans. y = (sin* 1 #)" + c, sin" 1 x + c v 



THE TWO FIRST INTEGRALS. 31 



* Prob. 56. Solve = . Ans. y = - + cjc\ 

dx x x 

v Prob. 57. Solve -^ = cfx tfy. 

Ans. c?x b*y A sin for -f- .5 cos for. 
vprob. 58. Solve y + ' = i. Ans. / = ** + ^ + c v 



ART. 14. THE Two FIRST INTEGRALS. 

We have seen in the preceding article that the complete 
integral of an equation of the second order is a relation be- 
tween x, y and two constants c t and c 9 . Conversely, any rela- 
tion between x, y and two arbitrary constants may be regarded 
as a primitive, from which a differential equation free from both 
arbitrary constants can be obtained. The process consists in 
first. obtaining, as in Art. 3, a differential equation of the first 
order independent of one of the constants, say c 9 , that is, a rela- 
tion between x, y,p and <:, , and then in like manner eliminating 
r, from the derivative of this equation. The result is the equa- 
tion of the second order or relation between x, y, p and q (q 
denoting the second derivative), of which the original equation 
is the complete primitive, the equation of the first order being 
the first integral in which c l is the constant of integration. It 
is obvious that we can, in like manner, obtain from the primi- 
tive a relation between x, y, p and c 9 , which will also be a first 
integral of the differential equation. Thus, to a given form of 
the primitive or complete integral there corresponds two first 
integrals. 

Geometrically the complete integral represents a doubly 
infinite system of curves, obtained by varying the values of c t 
and of independently. If we regard c t as fixed and c t as 
arbitrary, we select ffom that system a certain singly infinite 
system ; the first integral containing c, is the differential equa- 
tion of this system, which, as explained in Art. 2, is a relation 
between the coordinates of a moving point and the direction 
of its motion common to all the curves of the system. But 



DIFFERENTIAL EQUATIONS. 



the equation of the second order expresses a property involv- 
ing curvature as well as direction of path, and this property 
being independent of c l is common to all the systems corre- 
sponding to different values of c lt that is, to the entire doubly 
infinite system. A moving point, satisfying this equation, 
may have any position and move in any direction, provided its 
path has the proper curvature as determined by the value of q 
derived from the equation, when the selected values of x, y 
and/ have been substituted therein.* 

For example, equation (7) of the preceding article repre- 
sents an ellipse having its center at the origin and touching 
the lines x = I, as in the diagram ; c 1 is the ordinate of the 
point -of contact with x = i, and c 9 that of the point in which 
the ellipse cuts the axis of y. If we regard ^, as fixed and c, 
as arbitrary, the equation represents the system of ellipses 
touching the two lines at fixed points, and equation (6) is the 
differential equation of this system. In 
like manner, if , is fixed and c 1 arbitrary, 
equation (7) represents a system of ellipses 
cutting the axis of y in fixed points 
and touching the lines x= i. The 
corresponding differential equation will be 
found to be 




Finally, the equation of the second order, independent of , 
and [(5) of the preceding article] is the equation of the 
doubly infinite system of conies f with center at the origin, 
and touching the fixed lines x i,. 

* If the equation is of the second or higher degree in q, the condition for 
equal roots is a relation between x, y and/, which may be found to satisfy the 
given equation. If it does, it represents a system of singular solutions; each 
of the curves of this system, at each of its points, not only touches but osculates 
with a particular integral curve. It is to be remembered that a singular solu- 
tion of a first integral is not generally a solution of the given differential equa- 
tion; for it represents a curve which simply touches but does not osculate a set 
of curves belonging to the doubly infinite system. 

f Including hyperbolas corresponding to imaginary values of c*. 



THE TWO FIRST INTEGRALS. 33 

But, starting from the differential equation of second order, 
we may find other first integrals than those above which corre- 
spond to , and a . For instance, if equation (5) be multiplied 
by/, it becomes 



which is also an exact equation, giving the first integral 



in which c t is a new constant of integration. 

Whenever two first integrals have thus been found inde- 
pendently, the elimination of / between them gives the com- 
plete integral without further integration.* Thus the result 
of eliminating p between this last equation and the first inte- 
gral containing c l [equation (6), Art. 13] is 

/ -2c,xy + cfx 1 = C? - c t \ 

which is therefore another form of the complete integral. It 
is obvious from the first integral above that c t is the maximum 
value of y, so that it is the differential equation of the system 
of ellipse inscribed in the rectangle drawn in the diagram. A 
comparison of the two forms of the complete integral shows 
that the relation between the constants is c* = c? -j- c*. 

If a first integral be solved for the constant, that is, put in 
the form <j>(x, y, p) = c, the constant will disappear on differ- 
entiation, and the result will be the given equation of second 
order multiplied, in general, by an integrating factor. We can 
thus find any number of integrating factors of an equation 
already solved, and these may suggest the integrating factors 
of more general equations, as illustrated in Prob. 59 below. 

* The principle of this method has already been applied in Art. 10 to the 
solution of certain equations of the first order; the process consisted of forming 
the equation of the second order of which the given equation is a first integral 
(but with a particular value of the constant), then finding another first integral 
and deriving the complete integral by elimination of /. 



31 DIFFERENTIAL EQUATIONS. 

Prob. 59. Solve the equation y + c?y = o in the form 

y = A cos ax + B sin 0.x; 

and show that the corresponding integrating factors are also inte- 
grating factors of the equation 



where X is any function of x; and thence derive the integral of this 
equation. 

/* /* 

Ans. <y sin ax I cos ax . Xdx cos ax I sin fl.r . Xdx. 

Prob. 60. Find the rectangular and also the polar differential 
^equation of all circles passing through the origin. 



ART. 15. LINEAR EQUATIONS. 

A linear differential equation of any order is an equation of 
the first degree with respect to the dependent variable y and 
each of its derivatives, that is, an equation of the form 



where the coefficients />,... P n and the second member X are 
functions of the independent variable only. 

The solution of a linear equation is always supposed to be 
in the f orm y =f(x)\ and if j, is a function which satisfies the 
equation, it is customary to speak of the function j,, rather than 
of the equation y = jj/,, as an "integral" of the linear equa- 
tion. The general solution of the linear equation of the first 
order has been given in Art. 6. For orders higher than the 
first the general expression for the integrals cannot be effected 
by means of the ordinary functional symbols and the integral 
sign, as was done for the first order in Art. 6. 

The solution of equation (i) depends upon that of 



LINEAR EQUATIONS. 36 

The complete integral of this equation will contain arbi- 
trary constants, and the mode in which these enter the expres- 
sion for y is readily inferred from the form of the equation. 
For let y l be an integral, and c t an arbitrary constant ; the re- 
sult of putting y = cjf 1 in equation (2) is ^, times the result of 
putting y = y l ; that is, it is zero ; therefore c l y l is an integral. 
So too, if y t is an integral, j/ a is an integral ; and obviously 
also c l y l -\- c^y t is an integral. Thus, if n distinct integrals/,, 
y t ,. . . y n can be found, 

y = Wi + w* + - + c y (3) 

will satisfy the equation, and, containing, as it does, the proper 
number of constants, will be the complete integral. 

Consider now equation (i); let Fbe a particular integral of 
it, and denote by u the second member of equation (3), which 
is the complete integral when X = o. If 

y=Y-\-u (4) 

be substituted in equation (i), the result will be the sum of the 
results of putting y = Fand of putting y = u ; the first of 
these results will be X, because Fis an integral of equation (i), 
and the second will be zero because u is an integral of equa- 
tion (2). Hence equation (4) expresses an integral of (i); and 
since it contains the n arbitrary constants of equation (3), it 
is the complete integral of equation (i). With reference to 
this equation F is called " the particular integral," and u is 
called "the complementary function." The particular integral 
contains no arbitrary constant, and any two particular integrals 
may differ by any multiple of a term belonging to the comple- 
mentary function. 

If one term of the complementary function of a linear 
equation of the second order be known, the complete solution 
can be found. For let y l be the known term ; then, if y = yp 
be substituted in the first member, the coefficient of v in the 
result will be the same as if v were a constant : it will there- 
fore be zero, and v being absent, the result will be a linear equa- 
tion of the first order for v', the first derivative of v. Under 



36 DIFFERENTIAL EQUATIONS. 

the same circumstances the order of any linear equation can 
in like manner be reduced by unity. 

A very simple relation exists between the coefficients of an 
exact linear equation. Taking, for example, the equation of 
the second order, and indicating derivatives by accents, if 



is exact, the first term of the integral will be P^y' Subtracting 
the derivative of this from the first member, the remainder is 
(/>, /V)y + P,y. The second term of the integral must 
therefore be (P l P ')y ; subtracting the derivative of this ex- 
pression, the remainder, (P t />/ -j- P<>"}y, must vanish. Hence 
P 9 PI -j- P " = o is the criterion for the exactness of the 
given equation. A similar result obviously extends to equa- 
tions of higher orders. 

V Prob. 61. Solve x (3 + x) -\- $y = o, noticing that e* is 

an integral. Ans. y c^ + c a (x a + 3^ -f 6.v + 6. 

v Prob. 62. Solve (x* x)-^-. -4- z(2x 4- i)-f- + 2y = o. 

dx ax 

Ans. (^ *) b y = f i(- a;4 ~~ 6' ra 4~ 2Jf -J 4x 3 log ^) + <r a ^: 3 . 

d*y . O d*y a dv 

v Prob. 63. Solve^rn + cos "^z ~ 2 sin P-^ y cos c/ = sin 2#. 
ac/ f at/ 

Ans. y e- sin '^ ^ sin *(^ + 



ART. 16. LINEAR EQUATIONS WITH CONSTANT 
COEFFICIENTS. 

The linear equation with constant coefficients and second 
member zero may be written in the form 

J*-*y + . . . + A n y = o, (i) 



7 Jt 

in which D stands for the operator -j-, D* for ^-,, etc., so that 

D" indicates that the operator is to be applied n times. Then, 
since ZV* = me mx , D*e mx = m*e mx , etc., it is evident that if 



LINEAR EQUATIONS, CONSTANT COEFFICIENTS. 37 

y e mx be substituted in equation (l), the result after rejecting 
the factor e* will be 

Aftf + A^"- 1 + . . . + A H = o. (2) 

Hence, if m satisfies equation (2), e mx is an integral of equation 
(i) ; and if m lt m^ . . . m n are n distinct roots of equation (2), 
the complete integral of equation (i) will be 

y = c^ x + /"* + . . . + c n e" n *. (3) 

For example, if the given equation is 

dy 



the equation to determine m is 

n? m 2 = o, 

of which the roots are m l = 2, m t = i; therefore the in- 

tegral is 

y = cS* + c,e-\ 

The general equation (i) may be written in the symbolic 
form f(D) .y = o, in which / denotes a rational integral func- 
tion. Then equation (2) is f(m) = o, and, just as this last 
equation is equivalent to 

(m m^(m m t ) . . . (m m n ) = o, (4) 

so the symbolic equation f(D) . y = o may be written 

(D - m t )(D - m,) ..."(/>- m n }y = o. (5) 

This form of the equation shows that it is satisfied by each of 
the quantities which satisfy the separate equations 

(D - m,}y = o, (D m^y = o...(D m n )y = o ; (6) 
that is to say, by the separate terms of the complete integral. 

If two of the roots of equation (2) are equal, say to m lt two 
of the equations (6) become identical, and to obtain the full 
number of integrals we must find two terms corresponding to 

the equation 

(D-m l )y = o; (7) 

in other words, the complete integral of this equation of which 
^ = e n ^ is known to be one integral. For this purpose we 



38 DIFFERENTIAL EQUATIONS. 

put, as explained in the preceding article, y =7,^. By differen- 
tiation, Dy = De m \ x v = e m ^ x (m.y -J- Dv] ; therefore 

(D m^f^v = e m *Dv. (8) 

In like manner we find 

(D m^e m ^ x v = e m i*D*v. (9) 

Thus equation (7) is transformed to D*v = o, of which the 
complete integral is v = c,x -{-c t ; hence that of equation (7) is 

y = ev(c l x+Ct). (io> 

These are therefore the two terms corresponding to the squared 
factor (D m,Y in f(D}y = o. 

It is evident that, in a similar manner, the three terms 
corresponding to a case of three equal roots can be shown to- 
be c m ^(c^ -\- CyX -(- c a ), and so on. 

The pair of terms corresponding to a pair of imaginary- 
roots, say m l = a -\- ifi, m t a i/3, take the imaginary form 



Separating the real and imaginary parts of &* and e-#*, and 
changing the constants, the expression becomes 

e ax (A cos fix-\-B sin fix}. (\ i) 

For a multiple' pair of imaginary roots the constants A and 
B must be replaced by polynomials as above shown in the case 
of real roots. 

When the second member of the equation with constant 
coefficients is a function of X, the particular integral can also 
be made to depend upon the solution of linear equations of 
the first order. In accordance with the symbolic notation 
introduced above, the solution of the equation 

JL_ ay = X , or (D - a}y = x (12) 

is denoted by y = (D a)~ l X, so that, solving equation (12), 
we have 

D^=-a X = r/-T-r (13) 

as the value of the inverse symbol whose meaning is " that 



LINEAR EQUATIONS, CONSTANT COEFFICIENTS. - 39 

function of x which is converted to X by the direct operation 
expressed by the symbol D a" Taking the most convenient 
special value of the indefinite integral in equation (13), it gives 
the particular integral of equation (12). In like manner, the par- 
ticular integral of f(D)y = X is denoted by the inverse symbol 

fi-fi-X. Now, with the notation employed above, the symbolic 
J\ i 

fraction may be decomposed into partial fractions with constant 

numerators thus: 

N 



in which each term is to be evaluated by equation (13), and 
may be regarded (by virtue of the constant involved in the 
indefinite integral) as containing one term of the complement- 
ary function. For example, the complete solution of the 
equation 



is thus found to be 
y = 

When X is a power of x the particular integral may be 
found as follows, more expeditiously than by the evaluation of 
the integrals in the general solution. For example, if X = x* 
the particular integral in this example may be evaluated by 
development of the inverse symbol, thus : 

_ i _ _ _! 

y ~ D*-D~2 X ~ 2 



* The validity of this equation depends upon the fact that the operations 
expressed in the second member of 

f(D) = (D - mi )(D _,) + ...+(/>_,,) 

are commutative, hence the process of verification is the same as if the equation 
were an algebraic identity. This general solution was published by Boole in 
the Cambridge Math. Journal, First Series, vol. n, p. 114. It had, however, 
been previously published by Lobatto, Theorie des Characteristiques, Amster- 
dam, 1837. 



40 DIFFERENTIAL EQUATIONS. 

The form of the operand shows that, in this case, it is only 
necessary to carry the development as far as the term contain- 
ing D\ 

For other symbolic methods applicable to special forms of 
X we must refer to the standard treatises on this subject. 

d*y dy 
Prob. 64. Solve JH-< 3-7 + y . 

^dx dx 

Ans. y = <**(Ax + ) + c<r*. 
' Prob. 65. Show that 



and that ~ sin (ax + ^) = T sin (ax + ft). 



Prob. 66. Solve (Z> 5 + i )y = e* + sin zx + sin x. (Compare 
Prob. 59, Art. 14.) 

Ans. y = A sin x -f- B cos x -\- %e* -J sin zx $x cos x. 

ART. 17. HOMOGENEOUS LINEAR EQUATIONS. 
The linear differential equation 



in which A , A^ etc., are constants, is called the "homogene- 
ous linear equation." It bears the same relation to x m that 
the equation with constant coefficients does to e mx . Thus, if 
yx m be substituted in this equation, the factor x m will divide 
out from the result, giving an equation for determining m, 
and the n roots of this equation will in general determine the 
n terms of the complete integral. For example, if in the 

equation 

jd*y . dy 
*'-4 + 2*/- - 2y - o 
da? dx 

we put y = x m , the result is m(m i) + 2m 2 o, or 
(m i)(m -}- 2) = o. , ^ 

The roots of this equation are m 1 = I and m 9 = 2. 
Hence y = c^x -j- CyX~* 

is the complete integral. 

Equation (i) might in fact have been reduced to the form 
with constant coefficients by changing the independent vari- 



HOMOGENEOUS LINEAR EQUATIONS. 41 

able to 0, where x = e , or 6 = log x. We may therefore at 
once infer from the results established in the preceding article 
that the terms corresponding to a pair of equal roots are of the 

form 

(c l + c t log x)x m , (2) 

and also that the terms corresponding to a pair of imaginary 
roots, a ifi, are 

x*[A cos (/3 log x) + B sin (ft log *)]. (3) 

The analogy between the two classes of linear equations 
considered in this and the preceding article is more clearly 
seen when a single symbol $= xD is used for the operation of 
taking the derivative and then multiplying by x, so that 
$x m = mx m . It is to be noticed that the operation x^D 1 is not 
the same as & or xDxD, because the operations of taking the 
derivative and multiplying by a variable are not "commu- 
tative," that is, their order is not indifferent. We have, on the 
contrary, x^D* = 8(8 i) ; then the equation given above, 

which is 

(x*D* + 2xD 2)y = o, 
becomes 

[8(8 i) + 28 2]j = o, or (8-1X8 + 2)^ = 0, 

the function of 8 produced being the same as the function of 
m which is equated to o in finding the values of m. 

A linear equation of which the first member is homoge- 
neous and the second member a function of x may be reduced 
to the form 

A$).y = x- (4) 

The particular integral may, as in the preceding article (see 
eq. (14)), be separated into parts each of which depends upon 
the solution of a linear equation of the first order. Thus, 
solving the equation 



-ay = X, or (8 - a)y = X, (5) 



we find 



X=x" Cx- a - l Xdx. (6) 

a v 



8 
The more expeditious method which may be employed 



DIFFERENTIAL EQUATIONS. 



when X is a power of x is illustrated in the following example : 



d, v /z"i/ 

Given x* 2-f- = *'. The first member becomes homo- 
dx ax 

geneous when multiplied by x, and the reduced equation is 

( 8 _ 3$'jjy X \ 

The roots of /($) =o are 3 and the double root zero, hence 
the complementary function is cj? -f- c ., -f- c 3 log x. Since in 
general f($)x r f(r)x r , we infer that in operating upon x* we 
may put $ = 3. This gives for the particular integral 
i i ^ _ i i ^ 

but fails with respect to the factor $ 3.* We therefore 
now fall back upon equation (6), which gives 

JT- x 3 = x* / x~ l dx = x* log x. 
The complete integral therefore is 

y = 



dy 

Prob. 67. Solve zx -~ 4- 3^-7 $y = x*. 

ax ax 

Ans. y = c 1 x + 

Prob. 68. Solve (*'>' + 3 ^Z>' + D)y = -. 

oc 

Ans. y = c, + <r, log x + <r,(log ^) a 



ART. 18. SOLUTIONS IN INFINITE SERIES. 

We proceed in this article to illustrate the method by 
which the integrals of a linear equation whose coefficients are 
algebraic functions of x may be developed in series whose 
terms are powers of x. For this purpose let us take the 
equation 

' ' 



* The failure occurs because x 3 is a term of the complementary function 
having an indeterminate coefficient; accordingly the new term is of the same 
form as the second term necessary when 3 is a double root, but of course with 
a determinate coefficient. 



SOLUTIONS IN INFINITE SERIES. 43- 

which is known as " Bessel's Equation," and serves to define- 
the "Besselian Functions." 

If in the first member of this equation we substitute (or y 
the single term Ax m the result is 

A(m* - ri l }x m + Ax m +\ (2) 

the first term coming from the homogeneous terms of the 
equation and the second from the term x*y which is of higher 
degree. If this last term did not exist the equation would be 
satisfied by the assumed value of y, if m were determined so as 
to make the first term vanish, that is, in this case, by Ax n or 
Bx~ n . Now these are the first terms of two series each of 
which satisfies the equation. For, if we add to the value of y 
a term containing x m+2 , thus/ = A Q x m -j- A^" 1 ^ 2 , the new term- 
will give rise, in the result of substitution, to terms containing. 
x m+2 and x m+4 respectively, and it will be possible so to take 
A i that the entire coefficient of x m+ * shall vanish*. In like 
manner the proper determination of a third term makes the 
coefficient of x mJr * in the result of substitution vanish, and so 
on. We therefore at once assume 



= A,x' + A, x m + 2 -\- A t x m + 4 + . . . , (3) 

in which r has all integral values from o to oo. Substituting 
in equation (i) 

2[{(m + 2/) 9 - n 9 }A^ m + 2r -\- ^X* +2(r+1) ] = o. (4) 

The coefficient of each power of x in this equation must sep- 
arately vanish ; hence, taking the coefficient of x m+2r , we have 

[(m + 2 ry-n>]A r +A r _ I =o. (5) 

When r = o, this reduces to m* n* = o, which determines 
the values of m, and for other values of r it gives 

~ (m + 2r + n)(m -\-2r- n) Ar ~ 1 ' 
the relation between any two successive coefficients. 

For the first value of m, namely n, this relation becomes 

A __ !_ . A 
"'- ' -" 



4-i DIFFERENTIAL EQUATIONS. 

whence, determining the successive coefficients in equation (3), 
the first integral of the equation is 

- -^ -, + wlw 



In like manner, the other integral is found to be 



r-jj - .. .J. (7) 

. . -, (8) 



and the complete integral is 7 = A y l -J- 

This example illustrates a special case which may arise in 
this form of solution. If n is a positive integer, the second 
series will contain infinite coefficients. For example, if n == 2, 
the third coefficient, or B v is infinite, unless we take B = o, in 
which case B^ is indeterminate and we have a repetition of the 
solution y r This will always occur when the same powers of 
x occur in the two series, including, of course, the case in which 
m has equal roots. For the mode of obtaining a new integral 
in such cases the complete treatises must be referred to.f 

It will be noticed that the simplicity of the relation between 
consecutive coefficients in this example is due to the fact that 
equation (i) contained but two groups of terms producing 
different powers of x, when Ax m is substituted for y as in ex- 
pression (2). The group containing the second derivative 
necessarily gives rise to a coefficient of the second degree in 
in, and from it we obtained two values of m. Moreover, be- 
cause the other group was of a degree higher by two units, the 
assumed series was an ascending one, proceeding by powers 
of x\ 

* The Besselian function of the wth order usually denoted byy is the value 

, of y\ above, divided by 2"! if n is a positive integer, or generally by 2 n r(n-\-i). 

For a complete discussion of these functions see Lommel's Studien liber die 

Bessel'schen Functionen, Leipzig, 1868; Todhunter's Treatise on Laplace's, 

Lame's and Bessel's Functions, London, 1875, etc - 

f A solution of the kind referred to contains as one term the product of the 
regular solution and log x, and is sometimes called a " logarithmic solution." 
See also American Journal of Mathematics, Vol. XI, p. 37. In the case of 
Bessel's equation, the logarithmic solution is the "Besselian Function of the 
iecond kind." 



SOLUTIONS IN INFINITE SERIES. 45 

In the following example, 



there are also two such groups of terms, and their difference 
of degree shows that the series must ascend by simple powers. 
We assume therefore at once 



The result of substitution is 



- l ']= o. 



Equating to zero the coefficient of x" lJrr ~ 2 , 

(m -\- r + i)(m + r 2)A r + a(m + r i)A r . t = o, (12) 
which, when r = o, gives 

(;//+!)(* 2)/4.=o, (13) 

and when r > o, 

m-\-r I 

A r ^7 - i - i - \7 - i --- f^*r 1" (14) 

(in -\- r -{- i)(m -\-r~-2) 

The roots of equation (13) are m-=.2 and m = i; taking 
nt=2, the relation (14) becomes 

A i r+l A 

(r+*Y 
vh< nee the first integral is 



Taking the second value w = i, equation (14) gives 

r ~ 2 



\- 

3) 

, = 
gral is the finite expression 



whence B, = -- ^ , and ^, = o*; therefore the second inte~ 



* Bt would take the indeterminate form, and if we suppose it to have a finite 
value, the rest of the series is equivalent to B^y\, reproducing the first integral. 



46 DIFFERENTIAL EQUATIONS. 

When the coefficient of the term of highest degree in the 
result of substitution, such as equation (11), contains m, it is 
possible to obtain a solution in descending powers of x. In 
this case, m occurring only in the first degree, but one such 
solution can be found; it would be identical with the finite 
integral (16). In the general case there will be two such solu- 
tions, and they will be convergent for values of x greater than 
unity, while the ascending series will converge for values less 
than unity.* 

When the second member of the equation is a power of x, 
the particular integral can be determined in the form of a series 
in a similar manner. For example, suppose the second mem- 
ber of equation (9) to have been x*. Then, making the sub- 
stitution as before, we have the same relation between consecu- 
tive coefficients; but when r = o, instead of equation (13) we 
have 

(m 4- i)(m 2)A x m ~ 2 = x 

to determine the initial term of the series. This gives m = 2$ 
and A -f ; hence, putting m = in equation (14), we find for 
the particular integral f 



7 9.3 9.11.3.5 

A linear equation remains linear for two important classes 
of transformations ; first, when the independent variable is 
changed to any function of x, and second, when for y we put 
vf(x). As an example of the latter, let y = e~ ax v be substituted 
in equation (9) above. After rejecting the factor e'**, the 
result is 

d*v dv 2v _ 
dx* dx x* 

Since this differs from the given equation only in the sign 

*When there are two groups of terms, the integrals are expressible in terms 
of Gauss's " Hypergeometric Series." 

f If the second member is a term of the complementary function (for ex- 
ample, in this case, if it is any integral power of x), the particular integral will 
take the logarithmic form referred to in the foot-note on p. 346. 



SYSTEMS OF DIFFERENTIAL EQUATIONS. 47 

of a, we infer from equation (16) that it has the finite integral 
v = - -J . Hence the complete integral of equation (9) can 

X 

be written in the form 

xy <r,(2 ax) -f c^e- ax (2 -f- ax). 

<Ty 
Prob. 69. Integrate in series the equation -T-J + ^ry = o. 

Ans. ^(.-I^+Lj**-- . . .)-HB(_J^+'^'_ . . .). 

^y . </y 
Prob. 70. Integrate in series x*. -, + x* + (x 2)y = o. 

Prob. 71. Derive for the equation of Prob. 70 the integral 
y 9 = e~*(x~ l + i + *)> an d find its relation to those found above. 



ART. 19. SYSTEMS OF DIFFERENTIAL EQUATIONS. 

It is shown in Art/ 12 that a determinate system of n differ- 
ential equations of the first order connecting n -J- I variables 
has for its complete solution as many integral equations con- 
necting the variables and also involving n constants of inte- 
gration. The result of eliminating n I variables would be a 
single relation between the remaining two variables containing 
in general the n constants. But the elimination may also be 
effected in the differential system, the result being in general 
an equation of the wth order of which the equation just men- 
tioned is the complete integral. For example, if there were 
two equations of the first order connecting the variables x and 
y with the independent variable /, by differentiating each we 
should have four equations from which to eliminate one vari- 
able, say y, and its two derivatives * with respect to /, leaving 
a single equation of the second order between x and /. 

It is easy to see that the same conclusions hold if some of 
the given equations are of higher order, except that the order 
of the result will be correspondingly higher, its index being in 

* In general, there would be n* equations from which to eliminate i 
variables and n derivatives of each, that is, (n i)( -f- i) = n* I quantities 
leaving a single equation of the wth order. 



48 DIFFERENTIAL EQUATIONS. 

general the sum of the indices of the orders of the given equa- 
tions. The method is particularly applicable to linear equa- 
tions with constant coefficients, since we have a general method 
of solution for the final result. Using the symbolic notation, 
the differentiations are performed simply by multiplying by 
the symbol D, and therefore the whole elimination is of exactly 
the same form as if the equations were algebraic. For ex- 
ample, the system 

cFy dx _ dx dy 

when written symbolically, is 
(2D* 4)y Dx = 2t, 
whence, eliminating x, 

2t 



2 D*-4 D 
2D 4D~ y ~ 



o 



which reduces to 

(D-i) 
Integrating, 

y = (A + Bty + Ce~* - \t, 

the particular integral being found by symbolic development, 
as explained at the end of Art. 16. 

The value of x found in like manner is 

x = (A' + B'ty + C'e-V - f 

The complementary function, depending solely upon the deter- 
minant of the first members,* is necessarily of the same form 
as that for y, but involves a new set of constants. The re- 
lations betv/een the constants is found by substituting the 
values of x and y in one of the given equations, and equating 
to zero in the resulting identity the coefficients of the several 
terms of the complementary function. In the present ex- 
ample we should thus find the value of x, in terms of A, B, 
and C, to be 



* The index of the degree in D of this determinant is that of the order of 
the final equation ; it is not necessarily the sum of the indices of the orders of 
the given equations, but cannot exceed this sum. 



SYSTEMS OF DIFFERENTIAL EQUATIONS. 49" 

In general, the solution of a system of differential equations 
depends upon our ability to combine them in such a way as 
to form exact equations. For example, from the dynamical 
system 



~dJ- ' d?~ ' d? - 

where X, Y, Z are functions of x, y, and 2, but not of t t 
we form the equation 

dx ,dx . dy jdz . dz ,dz , , ,,, , 
- d -- r- -^-d-~ + -rd Xdx -4- Ydy -\-Zdz. 
dt dt '<#<#' dt dt 

The first member is an exact differential, and we know that for 
a conservative field of force the second member is also exact, 
that is, it is the differential of a function U of x> y, and z. 
The integral 



is that first integral of the system (i) which is known as the 
equation of energy for the unit mass. 

Just as in Art. 13 an equation of the second order was re- 
garded as equivalent to two equations of the first order, so the 
system (i) in connection with the equation defining the resolved 
velocities forms a system of six equations of the first order, of 
which system equation (2) is an " integral " in the sense ex- 
plained in Art. 12. 

Prob. 72. Solve the equations - = = dt as a system im- 

my mx 

ear in /. Ans. x A cosmt-\-smmt,y=A sinmtJB cosmt. 

Prob. 73. Solve the system -5 -- 1- *y = e*. -f- -4- z = o. 

dx dx 

Ans. y - Ae nx + Be~ nx + -r . z nAe nx -\- nEe'** -- ^ . 

n t n i 

Prob. 74. Find for the system - = x<t>(x,y), -^ = y<f>(x,y} 

a first integral independent of the function <f>. 

dy dx 



50 DIFFERENTIAL EQUATIONS. 

Prob. 75. The approximate equations for the horizontal motion 
of a pendulum, when the earth's rotation is taken into account, are 

d*x dy , gx d*y . dx , gy 

2f-> r i +i r = > i + * r is+i = < 

show that both x and y are of the form 

A cos ,/ -f- -B sin ,/ + C cos nj -f- D sin #, 

ART. 20. FIRST ORDER AND DEGREE WITH THREE 
VARIABLES. 

The equation of the first order and degree between three 
variables x, y and z may be written 

Pdx + Qdy + Rdz = o, (i) 

where P, Q and R are functions of x, y and z. When this 
equation is exact, P, Q and R are the partial derivatives of 
some function u, of x, y and z ; and we derive, as in Art. 4, 

'dP = 3Q dQ = d_ -dR = -dP_ ,. 

-dy d*' dz ~ 'dy' d* dz 

for the conditions of exactness. In the case of two variables, 
when the equation is not exact integrating factors always exist; 
but in this case, there is not always a factor // such that j*P, 
pQ and pR (put in place of P, Q, and R) will satisfy all three 
of the conditions (2). It is easily shown that for this purpose 
the relation 



must exist between the given values of P, Q, and R. This is 
therefore the " condition of integrability " of equation (i).* 

When this condition is fulfilled equation (i) may be inte- 
grated by first supposing one variable, say z, to be constant. 
Thus, integrating Pdx -f- Qdy = o, and supposing the constant 
of integration C to be a function of z, we obtain the integral, so 

* When there are more than three variables such a condition of integra- 
bility exists for each group of three variables, but these conditions are not all 
independent. Thus with four variables there are but three independent con- 
ditions. 



FIRST ORDER AND DEGREE, THREE VARIABLES. 51 

far as it depends upon x and y. Finally, by comparing the 
total differential of this result with the given equation we de- 
termine dC in terms of z and dz, and thence by integration the 
value of C. 

It may be noticed that when certain terms of an exact 
equation forms an exact differential, the remaining terms must 
also be exact. It follows that if one of the variables, say z 
can be completely separated from the other two (so that in 
equation (i) ^becomes a function of z only and P and Q func- 
tions of x and y, but not of z) the terms Pdx -f- Qdy must be 
thus rendered exact if the equation is integrable.* For example, 

zydx zxdy y*dz = o. 

is an integrable equation. Accordingly, dividing by y*2. which 
we notice separates the variable z from x and y, puts it in the 

exact form 

ydx xdy dz 



y 

of which the integral is x = y log cz. 

Regarding x, y and z as coordinates of a moving point, 
an integrable equation restricts the point to motion upon one 
of the surfaces belonging to the system of surfaces represented 
by the integral ; in other words, the point (x, y, z) moves in an 
arbitrary curve drawn on such a surface. Let us now consider 
in what way equation (i) restricts the motion of a point when 
it is not integrable. The direction cosines of a moving point 
are proportional to dx, dy, and dz; hence, denoting them by 
/, m and , the direction of motion of the point satisfying 
equation (i) must satisfy the condition 

Pl+Qm+Rn = o. (4) 

It is convenient to considernn this connection an auxiliary 
system of lines represented, as explained in Art. 12, by the 
simultaneous equations 

dx dy dz 

= = fe\ 

P Q R' 

*In fact for this case the condition (3) reduces to its last term, which ex- 
presses the exactness of Pdx -f- Qdy, 



52 DIFFERENTIAL EQUATIONS. 

The direction cosines of a point moving in one of the lines 
of this system are proportional to P, Q and R. Hence, de- 
noting them by A, ju, v, equation (4) gives 

A/ -j- pin -(- vn = o (6) 

for the relation between the directions of two moving points, 
whose paths intersect, subject respectively to equation (i) and 
to equations (5). The paths in question therefore intersect at 
right angles; therefore equation (i) simply restricts a point to 
move in a path which cuts orthogonally the lines of the auxili- 
ary system. 

Now, if there be a system of surfaces which cut the auxiliary 
lines orthogonally, the restriction just mentioned is completely 
expressed by the requirement that the line shall lie on one of 
these surfaces, the line being otherwise entirely arbitrary.. 
This is the case in which equation (i) is integrable.* 

On the other hand, when the equation is not integrable, the 
restriction can only be expressed by two equations involving 
an arbitrary function. Thus if we assume in advance one such 
relation, we know from Art. 12 that the given equation (i) 
together with the first derivative of the assumed relation forms 
a system admitting of solution in the form of two integrals- 
Both of these integrals will involve the assumed. function. For 
any particular value of that function we have a system of lines 
satisfying equation (i), and the arbitrary character of the func- 
tion makes the solution sufficiently general to include all lines 
which satisfy the equation.f 

Prob. 76. Show that the equation 

(mz ny)dx + (nx lz]dy + (ly mx}dz = o 
is integrable, and infer from the integral the character of the auxil- 

*It follows that, with respect to the system of lines represented by equations 
(5), equation (3) is the condition that the system shall admit of surfaces cutting 
them orthogonally. The lines of force in any field of conservative forces form 
such a system, the orthogonal surfaces being the equipotential surfaces. 

f So too there is an arbitrary element about the path of a point when the 
single equation to which it is subject is integrable, but this enters only into one 
of the two equations necessary to define the path. 



PARTIAL EQUATIONS, FIRST ORDEX OO 

iary lines. (Compare the illustrative example at the end of Art. 12.) 

Ans. nx Iz = C(ny mz). 

Prob. 77. Solve yz'dx z*dy e*dz o. Ans. yz = e*(i-\-cz). 
Prob. 78. Find the equation which in connection with^ .f(x) 
forms the solution of dz = aydx -f- bdy. 

Prob. 79. Show that a general solution of 

ydx (x z)(dy dz) 
is given by the equations 

yz=(t>(x), y = (x z) (/>'(*). 
(This is an example of " Monge's Solution.") 



ART. 21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST 
ORDER AND DEGREE. 

Let x denote an unknown function of the two independent 
variables x and y, and let 



denote its partial derivatives : a relation between one or both 
of these derivatives and the variables is called a " partial dif- 
ferential equation " of the first order! A value of z in terms of 
x and y which with its derivatives satisfies the equation, or a 
relation between x, y and z which makes z implicitly such a 
function, is a " particular integral." The most general equation 
of this kind is called the " general integral." 

If only one of the derivatives, say/, occurs, the equation 
may be solved as an ordinary differential equation. For if y is 
considered as a constant,/ becomes the ordinary derivative of 
z with respect to x\ therefore, if in the complete integral of 
the equation thus regarded we replace the constant of integra- 
tion by an arbitrary function of j, we shall have a relation 
which includes all particular integrals and has the greatest pos- 
sible generality. It will be found that, in like manner, when 
both p and q are present, the general integral involves an arbi- 
trary function, 

We proceed to give Lagrange's solution of the equation of 



54r DIFFERENTIAL EQUATIONS. 

the first order and degree, or " linear equation," which may be 
written in the form 

Pp+Qq = R, (I) 

P, Q and R denoting functions of x, y and z. Let u = a, in 
which u is a function of x, y and z, and #, a constant, be an 
integral of equation (i). Taking derivatives with respect to x 
and y respectively, we have 



and substitution of the values of / and q in equation (i) gives 
the symmetrical relation 



Consider now the system of simultaneous ordinary differ- 
ential equations 

dx_ _ dy__ dz_ 

~J>~~Q^~R (3) 

Let u = a be one of the integrals (see Art. 12) of this sys- 
tem. Taking its total differential, 

"du 3 9 

-^+-^ + --^=0: 

and since by equations (3) dx, dy and dz are proportional to />, 
<2 and 7?, we obtain by substitution 



which is identical with equation (2). It follows that every 
integral of the system (3) satisfies equation (i), and conversely, 
so that the general expression for the integrals of (3) will be 
the general integral of equation (i). 

Now let v == b be another integral of equations (3), so that 
v is also a function which satisfies equation (2). As explained 
in Art. 12, each of the equations u a, v = b is the equation 
of a surface passing through a singly infinite system of lines 
belonging to the doubly infinite system represented by equa- 
tions (3). What we require is the general expression for any 



PARTIAL EQUATIONS, FIRST ORDER. 



55 



surface passing through lines of the system (and intersecting 
none of them). It is evident that f(u, v) = f(a, b) = C is such 
an equation,* and accordingly f(u, v), where f is an arbitrary 
function, will be found to satisfy equation (2). Therefore, to 
solve equation (i), we find two independent integrals u = a, 
v = b of the auxiliary system (3), (sometimes called Lagrange's 
equations,) and then put 

= 00)> (4) 

an equation which is evidently equally general with/(#, v) = o. 

Conversely, it may be shown that any equation of the form 
(4), regarded as a primitive, gives rise to a definite partial 
differential equation of Lagrange's linear form. For, taking 
partial derivatives with respect to the independent variables 
x and j, we have 



3*^3-3- 



and eliminating (f>'(v) from these equations, the term contain- 
ing/^ vanishes, giving the result 

3 3; 



37 



P + 



(5) 



which is of the form Pp -f- Qq R.\ 

* Each line of the system is characterized by special values of a and b which 
we may call its coordinates, and the surface passes through those lines whose 
coordinates are connected by the perfectly arbitrary relation f(a, 6) = C. 

f These values of P, Q and R are known as the " Jacobians " of the pair 
of functions u, v with respect to the pairs of variables y, z ; z, x ; and x, y re- 
spectively. Owing to their analogy to the derivatives of a single function they 
are sometimes denoted thus : 

_ 3(, z/) 3(, 



3(.v, 



g _ d(u, v) 



The Jacobian vanishes if the functions u and v are not independent, that is 
to say, if can be expressed identically as a function of v. In like manner, 



56 DIFFERENTIAL EQUATIONS. 

As an illustration, let the given partial differential equa- 

tion be 

(mz ny]p -f- (nx lz)q = ly mx, (6) 

.for which Lagrange's Equations are 

dx dy dz 

_____^^__ * __ 17 > 

mz ny ~ nx Iz ~ ly mx' *'' 

These equations were solved at the end of Art. 12, the two 
integrals there found being 

Ix -j- my -\- nz = a and x* -\- y* -j- ^ = b*> (8) 

Hence in this case the system of " Lagrangean lines" con- 
sists of the entire system of circles having the straight line 



for axis. The general integral of equation (6) is then 

Ix -j- my + nz = (t>(x* -\- y* -f- z*), (10) 

which represents any surface passing through the circles just 
mentioned, that is, any surface of revolution of which (9) is the 
axis.* 

Lagrange's solution extends to the linear equation contain- 
ing n independent variables. Thus the equation being 



the auxiliary equations are 

dx\ dx^ _ _ dx n _ dz 

~I\'-~-^\~- : ^ = ^~' 

! - = o is the condition that (a function of x, y and 2) is expressible 

<H*, >, 2) 

identically as a function of u and v, that is to say, that = o shall be an in- 
tegral of Pp + Qq= R. 

* When the equation Pdx -f- Qdy + Rdz = o is integrable (as it is in the 
above example; see Prob. 76, Art. 20), its integral, which may be put in the form 

V = C, represents a singly infinite system of surfaces which the Lagrangean 
lines cut orthogonally ; therefore, in this case, the general integral may be de- 
fined as the general equation of the surfaces which cut orthogonally the system 

V = C. Conversely, starting with a given system V = C, u = J\v) is the gen- 
eral equation of the orthogonal surfaces, if u = a and v = b are integrals of 



COMPLETE AND GENERAL INTEGRALS. 5? 

and if u l = c lt u y = c a , . . . u n = c n are independent integrals, 
the most general solution is 

/(,, . . . ) = o, 
where /is an arbitrary function. 

^ 2 ^ 2T / y\ 

Prob. 80. Solve xz- \-yz~~ = xy. Ans. xy z* =/[ ). 
dx dy \y) 

Prob. 81. Solve (y + z)p + (z + x}q = x + y. 
Prob. 82. Solve (x +^)(/ q) z. 

Ans. (x-\-y) log 2 x =/(x-\-y). 
Prob. 83. Solve x(y z)p -\-y(z x)q = z(x y). 

Ans. x -\- y-\- z= f(xyz). 

ART. 22. COMPLETE AND GENERAL INTEGRALS. 

We have seen in the preceding article that an equation be- 
tween three variables containing an arbitrary function gives 
rise to a partial differential equation of the linear form. It 
follows that, when the equation is not linear in / and q, the 
general integral cannot be expressed by a single equation of 
the f^rm 0(, v) = o; it will, however, still be f^und to- depend 
upon a single arbitrary function. X . 

It therefore becomes necessary to consider an integral hav- 
ing as much generality as can be given by the presence of arbi- 
trary constants. Such an equation is called a " complete in- 
tegral " ; it contains two arbitrary constants (n arbitrary con- 
stants in the general case of n independent variables), because 
this is the number which can be eliminated from such an equa- 
tion, considered as a primitive, and its two derived equations. 
For example, if 

(*-a)' + o/-)' + *' = /P, 

a and b being regarded as arbitrary, be taken as the primitive, 
the derived equations are 

x a -\- zp = o, y b -f- zq = O, 
and the elimination of a and b gives the differential equation 

A/ +V +!)*. 

of which therefore the given equation is a complete integral. 



58 DIFFERENTIAL EQUATIONS. 

Geometrically, the complete integral represents a doubly in- 
finite system of surfaces ; in this case they are spherical sur- 
faces having a given radius and centers in the plane of xy. 

In general, a partial differential equation of the first order 
with two independent variables is of the form 

F(x, y, z, p, q) = O, (l) 

and a complete integral is of the form 

f(x, y, 2, a, b) = o. (2) 

In equation (i) suppose x, y and z to have special values, 
namely, the coordinates of a special point A ; the equation 
becomes a relation between p and q. Now consider any sur- 
face passing through A of which the equation is an integral of 
(i), or, as we may call it, a given "integral surface " passing 
through A. The tangent plane to this surface at A determines 
values of / and q which must satisfy the relation just men- 
tioned. Consider also those of the complete integral surfaces 
[equation (2)] which pass through A. They form a singly in- 
finite system whose-tangent planes at A have values of p and 
q which also satisfy the relation. There is obviously among 
them one which has the same value of /, and therefore also 
the same value of q, as the given integral. Thus there is one 
of the complete integral surfaces which touches at A the given 
integral surface. It follows that every integral surface (not in- 
cluded in the complete integral) must at every one of its points 
touch a surface included in the complete integral.* 

It is hence evident that every integral surface is the en- 
velope of a singly infinite system selected from the complete 
integral system. Thus, in the example at the beginning of 
this article, a right cylinder whose radius is k and whose axis 
lies in the plane of xy is an integral, because it is the envelope 

* Values of x, y, and z, determining a point, together with values of/ and q, 
determining the direction of a surface at that point, are said to constitute an 
"element of surface." The theorem shows that the complete integral is ' com- 
plete " in the sense of including all the surface elements which satisfy the differ- 
ential equation. The method of grouping the "consecutive" elements to form 
an integral surface is to a certain extent arbitrary. 



COMPLETE AND GENERAL INTEGRALS. 59 

of those among the spheres represented by the complete in- 
tegral whose centers are on the axis of the cylinder. If we 
make the center of the sphere describe an arbitrary curve in 
the plane of xy we shall have the general integral in this ex- 
ample. 

In general, if in equation (2) an arbitrary relation between 
a and b, such as b = </>(a), be established, the envelope of the 
singly infinite system of surfaces thus defined will represent 
the general integral. By the usual process, the equation of 
the envelope is the result of eliminating a between the two 
equations 

o 

f(x, y, z, a, 0(rt) ) = o, ~Tf( x i y> z > a > 0(*) ) = - (3) 



These two equations together determine a line, namely, the 
" ultimate intersection of two consecutive surfaces." Such 
lines are called the " characteristics " of the differential equa- 
tion. They are independent of any particular form of the 
complete integral, being in fact lines along which all integral 
surfaces which pass through them touch one another. In the 
illustrative example above they are equal circles with centers 
in the plane of xy and planes perpendicular to it.* 

The example also furnishes an instance of a " singular so- 
lution " analogous to those of ordinary differential equations. 

*The characteristics are to be regarded not merely as lines, but as " linear 
elements of surface," since they determine at each of their points the direction 
of the surfaces passing through them. Thus, in the illustration, they are cir- 
cles regarded as great-circle elements of a sphere, or as elements of a right 
cylinder, and may be likened to narrow hoops. They constitute in all cases a 
triply infinite system. The surfaces of a complete integral system contain them 
all, but they are differently grouped in different integral surfaces. 

If we arbitrarily select a curve in space there will in general be at each of 
its points but one characteristic through which the selected curve passes; that 
is, whose tangent plane contains the tangent to the selected curve. These char- 
acteristics (for all points of the curve) form an integral surface passing through 
the selected curve ; and it is the only one which passes through it unless it be 
itself a characteristic. Integral surfaces of a special kind result when the se- 
lected curve is reduced to a point. In the illustration these are the results of 
rotating the circle about a line parallel to the axis of z. 



60 DIFFERENTIAL EQUATIONS. 

For the planes z = k envelop the whole system of spheres 
represented by the complete integral, and indeed all the sur- 
faces included in the general integral. When a singular solu- 
tion exists it is included in the result of eliminating a and b 
from equation (2) and its derivatives with respect to a and b, 
that is, from 



but, as in the case of ordinary equations, this result may in- 
clude relations which are not solutions. 

Prob. 84. Derive a differential equation from the primitive 
2x + my -f- nz = a, where /, m, n are connected by the relation 
f + m* + *' = i. 

Prob. 85. Show that the singular solution of the equation 
found in Prob. 84 represents a sphere, that the characteristics con- 
sist of all the straight lines which touch this sphere, and that the 
general integral therefore represents all developable surfaces which 
touch the sphere. 

Prob. 86. Find the integral which results from taking in the 
general integral above /' -\-m* = cos" (a constant) for the arbitrary 
relation between the parameters. 

ART, 23. COMPLETE INTEGRAL FOR SPECIAL FORMS. 

A complete integral of the partial differential equation 

F(x,y,z,p,q) = o (i) 

contains two constants, a and b. If a be regarded as fixed and 
b as an arbitrary parameter, it is the equation of a singly in- 
finite system of surfaces, of which one can be found passing 
through any given point. The ordinary differential equation 
of this system, which will be independent of b, may be put in 

the form 

dz = pdx -\- qdy, (2) 

in which the coefficients/ and q are functions of the variables 
and the constant a. Now the form of equation (2) shows that 
these quantities are the partial derivatives of z, in an integral 
of equation (i); therefore they are values of p and q which 



COMPLETE INTEGRAL FOR SPECIAL FORMS. 61 

satisfy equation (i). Conversely, if values of/ and q in terms 
of the variables and a constant a which satisfy equation (i) are 
such as to make equation (2) the differential equation of a sys- 
tem of surfaces, these surfaces will be integrals. In other 
words, if we can find values of/ and q containing a constant a 
which satisfy equation (i) and make dz = pdx -f- qdy inte- 
grable, we can obtain by direct integration a complete inte- 
gral, the integration introducing a second constant. 

There are certain forms of equations for which such values 
of / and q are easily found. In particular there are forms in 
which / and q admit of constant values, and these obviously 
make equation (2) integrable. Thus, if the equation contains 
/ and q only, being of the form 

F(P* 4) = 0, (3) 

we may put p = a and q = b, provided 

F(a,t)=o. (4) 

Equation (2) thus becomes 

dz = adx -f- bdy, 
whence we have the complete integral 

z = ax+by-\ r c, (5) 

in which a and b are connected by the relation (4) so that a, b 
and c are equivalent to two arbitrary constants. 

In the next place, if the equation is of the form 

*=P* + W+f(P,q\ (6) 

which is analogous to Clairaut's form, Art. 10, constant values 
of p and q are again admissible if they satisfy 

z = ax + by+f(a,b\ (7) 

and this is itself the complete integral. For this equation is 
of the form z ax + by + c, and expresses in itself the rela- 
tions between the three constants. Problem 84 of the preced- 
ing article is an example of this form. 

In the third place, suppose the equation to be of the form 
F(z,p,q} = o, (g) 



62 DIFFERENTIAL EQUATIONS. 

in which neither x nor y appears explicitly. If we assume 
q = ap, p will be a function of z determined from 

F(z, p, ap] o, say / = (f>(z), (9) 

Then dz = pdx -\- qdy = o becomes dz (f)(z)(dx -|- ady), which is 
integrable, giving the complete integral 



A fourth case is that in which, while z does not explicitly 
occur, it is possible to separate x and / from y and q, thus put- 
ting the equation in the form 

/.(*>/) =/,(* ?) (II) 

If we assume each member of this equation equal to a con- 
stant #, we may determine/ and q in the forms 

/ = <&(>> a \ q = fad', a). (12) 

and dz = pdx + qdy takes an integrable form giving 

(13) 



It is frequently possible to reduce a given equation by trans- 
formation of the variables to one of the four forms considered 
in this article.* For example, the equation x*p* + y q* = z* 
may be written 



*The general method, due to Charpit, of finding a proper value olp consists 
of establishing, by means of the condition of integrability, a linear partial dif- 
ferential equation for/, of which we need only a particular integral. This may 
be any value of / taken from the auxiliary equations employed in Lagrange's 
process. See Boole, Differential Equations (London 1865), p. 336 ; also For- 
syth, Differential Equations (London 1885), p. 316, in which the auxiliary equa- 
tions are deduced in a more general and symmetrical form, involving both / 
and q. These equations are in fact the equations of the characteristics regarded 
as in the concluding note to the preceding article. Denoting the partial deriva- 
tives of F(x, y, z,p, q) by X, Y, Z, P, Q, they are 

dx _ dy dz dt> >/,/ 

' ~ ~ 



IP - Q Pp+Qq ~ X+ty ~ Y+Zq' 

See Jordan's Cours d'Analyse (Paris, 1887), vol. in, p. 318 ; Johnson's Differ- 
ential Equations (New York, 1889), p. 300. Any relation involving one or both 
the quantities /and q, combined with f=o, will furnish proper values of/ 



PARTIAL EQUATIONS, SECOND ORDER. 63 

whence, putting x' = log x, y' = log y, z' = log 2, it becomes 
/'* + ^' a == l > which is of the form F(p', q'} = o, equation (3). 
Hence the integral is given by equation (5) when a? -j- &* = i; 
it may therefore be written 

z' = x' cos a-\-y' sin a -f- c, 

and restoring x, y, and #, that of the given equation is 

z = cx co * a y* Q ". 

Prob. 87. Find a complete integral for/ 1 q* = i. 

Ans. 2 = .# sec or + j> tan a + . 
Prob. 88. Find the singular solution of z = px + gy +/? 

Ans. 2 = xy. 
Prob. 89. Solve by transformation q = zyp*. 

Ans. z = ax -\~ ofy* + ^ 
Prob. 90. Solve z(p*q*} = x y. 

Ans. 2? = (# + a)i + (y + )1 + b. 

Prob. 91. Show that the solution given for the form F(z,p, q ) o 
represents cylindrical surfaces, and that F(z, o, o) = o is a singular 
solution. 

Prob. 92. Deduce by the method quoted in the foot-note two 
complete integrals of pq = px -j- qy. 

f x Y 

Ans. 22 = f -f~ a y) + A an( i z== xy-\-y ^(x'' -}- a) -\- b. 

ART. 24. PARTIAL EQUATIONS OF SECOND ORDER. 

We have seen in the preceding articles that the general 
solution of a partial differential equation of the first order de- 
pends upon an arbitrary function ; although it is only when 
the equation is linear in / and q that it is expressible by a 
single equation. But in the case of higher orders no general 
account can be given of the nature of a solution. Moreover, 
when we consider the equations derivable from a primitive con- 
taining arbitrary functions, there is no correspondence between 
their number and the order of the equation. For example, if 

and q. Sometimes several such relations are readily found ; for example, for 
the equation z = pq we thus obtain the two complete integrals 

and 4* =/ 



64- DIFFERENTIAL EQUATIONS. 

the primitive with two independent variables contains two ar- 
bitrary functions, it is not generally possible to eliminate them 
and their derivatives from the primitive and its two derived 
equations of the first and three of the second order. 

Instead of a primitive containing two arbitrary functions, 
let us take an equation of the first order containing a single 
arbitrary function. This may be put in the form u = $(v\ 
u and v now denoting known functions of x, y, z, p, and q. 
(f>'(v) may now be eliminated from the two derived equations 
as in Art. 21. Denoting the second derivatives of z by 
9V 9** ,_9!f 

= a*" ~ a-ra/ ~ ay 

the result is found to be of the form 

Rr + Ss-\-Tt+ U(rt - /) = V, (i) 

in which R, S, T, U, and V are functions of x, y, z, p, and,^. 
With reference to the differential equation of the second order 
the equation u = </>(v) is called an " intermediate equation of 
the first order " : it is analogous to the first integral of an ordi- 
nary equation of the second order. It follows that an inter- 
mediate equation cannot exist unless the equation is of the 
form (i); moreover, there are two other conditions which 
must exist between the functions R, S, T, and U. 

In some simple cases an intermediate equation can be ob- 
tained by direct integration. Thus, if the equation contains 
derivatives with respect to one only of the variables, it may be 
treated as an ordinary differential equation of the second order, 
the constants being replaced by arbitrary functions of the 
other variable. Given, for example, the equation xr p = xy, 
which may be written 

xdp pdx = xy dx. 

This becomes exact with reference to x when divided by x*, 
and gives the intermediate equation 



A second integration (and change in the form of the arbitrary 
function) gives the general integral 
z = $yx* log x -f 



PARTIAL EQUATIONS, SECOND ORDER. 65 

Again, the equation p-\-r-\-s=i is already exact, and 
gives the intermediate equation 



which is of Lagrange's form. The auxiliary equations* are 

dz 

dx = dy = - -, 
x - z -f <t>(y) 

of which the first gives x y = a, and eliminating x from the 
second, its integral is of the form 

z = a -}- 0( y] 4" e ~ y b. 
Hence, putting b = $(a), we have for the final integral 



in which a further change is made in the form of the arbitrary 
function 0. 

Prob. 93. Solve / q = e* + e y . 

Ans. z = ye y c* + <p 



Prob. 94. Solve r+p* = y\ 

Ans. z = log [^00 - e-**] + 

Prob. 95. Solve /(j /) = x. 

Ans. 2 = (x +y) logj> + 0(*) + #(* 

Prob. 96. Solve /^ qr = o. Ans. ^c = 0(.y) + ^( 2 )- 

Prob. 97. Show that Monge's equations (see foot-note) give for 

Prob. 96 the intermediate integral p = (p(z) and hence derive the 

solution. 

* In Monge's method (for which the reader must be referred to the complete 
treatises) of finding an intermediate integral of 

Rr + Ss + Tt = V 
when one exists, the auxiliary equations 

Rdy* - Sdy dx + Tdo* = o, Rdp dy + Tdq dx = Vdx dy 
are established. These, in connection with 

dz = pdx -f- qdy, 

form an incomplete system of ordinary differential equations, between the five 
variables x, y, z, p, and q. But if it is possible to obtain two integrals of the 
system in the form = a, v = b t u = 0(z>) will be the intermediate integral. 
The first of the auxiliary equations is a quadratic giving two values for the ratio 
dy.dx. If these are distinct, and an intermediate integral can be found, for 
each, the values of p and q determined from them will make dz = pdx -(- qdy 
jntegrable, and give the general integral at once. 



6G DIFFERENTIAL EQUATIONS. 

Prob. 98. Derive by Monge's method for q*r 2pqs -\- p*t = o 
the intermediate integral/ = q <fi(z), and thence the general integral. 

Ans. y + x<f>(z) = 



ART. 25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS. 

Equations which are linear with respect to the dependent 
variable and its partial derivatives may be treated by a method 
analogous to that employed in the case of ordinary differential 
equations. We shall consider only the case of two independ- 
ent variables x and y, and put 

D- D' - 

" a*' ~ a/ 

so that the higher derivatives are denoted by the symbols D*, 
DD', D rt , D 3 , etc. Supposing further that the coefficients are 
constants, the equation may be written in the form 

f(D, D')z =. F(x, y\ (i) 

in which f denotes an algebraic function, or polynomial, of 
which the degree corresponds to the order of the differential 
equation. Understanding by an " integral" of this equation 
an explicit value of z in terms of x and y, it is obvious, as in 
Art. 1 5, that the sum of a particular integral and the general 

integral of 

AD, D'}z = o (2) 

will constitute an equally general solution of equation (i). It 
is, however, only when f(D, D') is a homogeneous function of D 
and D' that we can obtain a solution of equation (2) containing 
n arbitrary functions,* which solution is also the "comple- 
mentary function" for equation (i). 

Suppose then the equation to be of the form 



. 

and let us assume 2 = <p(y -f- mx), (4) 

* It is assumed that such a solution constitutes the general integral of an 
equation of the th order; for a primitive containing more than independent 
arbitrary functions cannot give rise by their elimination to an equation of the 
th order. 



LINEAR PARTIAL EQUATIONS. 67 

where m is a constant to be determined. From equation (4), 
Dz = m(p'(y -f- mx) and D 'z = <j>\y -j- mx}, so that Dz = mD'z, 
D*z = m*D'*z, DD'z = mD'*y, etc. Substituting in equation (3) 
and rejecting the factor D' n z or (f> (n) (y -\- mx), we have 

Ajn* + A,m n - 1 -f . . . + A n = o (5) 

for the determination of m. If ;,, t 9 , . . . m n are distinct roots 
of this equation, 

* = <t>i(y + m **) +<P*( y + m **} + + <i>n(y + m n x) (6) 

is the general integral of equation (3). 

For example, the general integral of ; --., = o is thus 

QX oy 

found to be z = (f)(y -\- x] -j- $(y *) Any expression of the 
form Axy + Bx -f- Cy -\- D is a particular integral ; accordingly 
it is expressible as the sum of certain functions of x -\- y and 
x y respectively. 

The homogeneous equation (3) may now be written sym- 
bolically in the form 

(D - mjy)(D - m,D'} ...(D- m n D'}z = o, (7) 

in which the several factors correspond to the several terms of 
the general integral. If- two of the roots of equation (5) are 
equal, say, to m lt the corresponding terms in equation (6) are 
equivalent to a single arbitrary function. To form the general 
integral we need an integral of 

(D - m.DJz - o (8) 

in addition to 0(j -)- m } x}. This will in fact be the solution of 

(D - m,D'}z = <f>(y + ,*); (9) 

for, if we operate with D m^D' upon both members of this 

equation, we obtain equation (8). Writing equation (9) in the 

form 

p m,q = (f>(y + mx\ 

Lagrange's equations are 



-V . f . k j 

1 0(7 + *!*) 

giving the integrals y -f- #*,.* = #, ^ = x<p(a) -}- b. Hence the 
integral of equation (9) is 

z = x(f)(y -|- m^x) -\- if)(y -\- ntji), (10) 



63 DIFFERENTIAL EQUATIONS. 

and regarding also as arbitrary, these are the two independ- 
ent terms corresponding to the pair of equal roots. 

If equation (5) has a pair of imaginary roots m = ^ zV, 
the corresponding terms of the integral take the form 



which when and ip are real functions contain imaginary 
terms. If we restrict ourselves to real integrals we cannot 
now say that there are two radically distinct classes of inte- 
grals ; but if any real function of y -(- JJLX ~\- ivx be put in the 
form X-\-iY, either of the real functions X or Y will be an 
integral of the equation. Given, for example, the equation 



of which the general integral is 

s = 0(7 + "0 

to obtain a real integral take either the real or the coefficient 
of the imaginary part of any real form of (f>(y -j- ix]. Thus, if 
0(/) = # we find e y cosx and e y sin;r, each of which is an 
integral (see Chap. VI, p. 245). 

As in the corresponding case of ordinary equations, the 
particular integral of equation (i) may be made to depend 
upon the solution of linear equations of the first order. The 

inverse symbol -=? ^^(x, y) in the equation corresponding 

to equation (14), Art. 16, denotes the value of z in 

(D mD'}z F(x, y) or p mq = F(x, y). (l 1} 
For this equation Lagrange's auxiliary equations give 
y -j- mx = a, z = / F(x, a mx)dx -f- b = F^(x, a) -f- b, 

and the general integral is 

mx). (12) 



The first term, which is the particular integral, may there- 
fore be found by subtracting mx from y in F(x, y}, inte- 



LINEAR PARTIAL EQUATIONS. 69 

grating with respect to x, and then adding mx to y in the 
result.* 

For certain forms of F(x, y) there exist more expeditious 
methods, of which we shall here only notice that which applies 
to the form F(ax -\- by). Since DF(ax -f- by) = aF'(ax -f by) 
and D'F(ax -f- by) = bF'(ax -\- by), it is readily inferred that, 
when f(D, D') is a homogeneous function of the wth degree, 

f(D,D')F(ax + by) = f(a, b)F*\ax + by). (13) 

That is, if t ax -\- by, the operation of f(D, D') on F(t) is 
equivalent to multiplication by/(a,b) and taking the th de- 
rivative, the final result being still a function of /. It follows 
that, conversely, the operation of the inverse symbol upon a 
function of t is equivalent to dividing by f(a, b) and integrating 
n times. Thus, 

ff. . .fPWT. (H) 



^ y 

When ax -(- by is a multiple of y -f- w,-^, where m^ is a root of 
equation (5), this method fails with respect to the correspond- 
ing symbolic factor, giving rise to an equation of the form (9), 
of which the solution is given in equation (10). Given, for ex- 
ample, the equation 

6*2 d*Z 6*2 



or (D D') (D + 2D')z = sin (x y) -f sin (x + y). 

The complementary function is <p(y + *") + $(y 2x). The 

part of the particular integral arising from sin (x y), in which 

a = i, b = i, is - - / / sin tdf = ~ sin (x y). That aris- 



* The symbolic form of this theorem is 



corresponding to equation (13), Art. 16. The symbol e^xD 1 here indicates the 
addition of mx to y in the operand. Accordingly, using the expanded form 
of the symbol, 

f\ 22 X2 
ffttxIJ f ( lA ~~" 1 1 | fttX ** ' ~~i ~^~ ~T~ ) f\ V) "^ f\ V ^~ fttX\ 

8y 2 ! <5y* 
the symbolic expression of Taylor's Theorem. 



70 DIFFERENTIAL EQUATIONS. 

ing from sin (x -\-y) which is of the form of a term in the com- 
plementary function is ^ jr, cos (x + j), which by equa- 

o 

tion (10) is \x cos (x -\-y). Hence the general integral of 
the given equation is 

* = 0( y + *) + $(y - 2*}+% sin (x -y)-\x cos (* + j).. 

If in the equation f(D, D')z = o the symbol f(D, D'), though 
not homogeneous with respect to D and D', can be separated 
into factors, the integral is still the sum of those corresponding 
to the several symbolic factors. The integral of a factor of 
the first degree is found by Lagrange's process; thus that of 

(D mD' a)z = o (15) 

is z = ^"0( y -\- mx). (16) 

But in the general case it is not possible to express the 
solution in a form involving arbitrary functions. Let us, how- 
ever, assume 

*=<***+*, (17) 

where c, h, and k are constants. Since De hx+ky = he kx + k:r 
and >'<**+**= k<* x + ky , substitution in f'(D,D'}z = o gives 
cf(fi, k)e hx ~ lrky = o. Hence we have a solution of the form (17) 
whenever h and k satisfy the relation 

Ak>k} = o, (18) 

c being altogether arbitrary. It is obvious that we may also 
write the more general solution 

z=2ce** +F <*, (19) 

where k = F(h) is derived from equation (18), and c and h admit 
of an infinite variety of arbitrary values. 

Again, since the difference of any two terms of the form 
fkx + FWy w jth different values of h is included in expression 
(19), we infer that the derivative of this expression with respect 
to h is also an integral, and in like manner the second and 
higher derivatives are integrals. 

For example, if the equation is 



LINEAR PARTIAL EQUATIONS. 71 

for which equation (18) is f? k o, we have classes of in- 
tegrals of the forms 



27)], e** + k (x -f 2/^) s + 6y(x -f- 



In particular, putting // = o we obtain the algebraic integrals 
Cl x y c&? + 27), ,(*' + 6*y), etc. 

The solution of a linear partial differential equation with 
variable coefficients may sometimes be effected by a change of 
the independent variables as illustrated in some of the exam- 
ples below. 

Prob. 99. Show that if m l is a triple root the corresponding. 
terms of the integral are o?<t>(y + m t x) -\- x$(y-{- w,#) 

B u 9 ' 2 9>j5 9 ^ 

Prob. 100. Solve 2^-^ -- 3^^ -- 2^ a = o. 

d* dxdy dy 



Prob. ici. Solve ^^ + 2 ^ + - 3 = ^. 



Ans. 2 = 0(^) -f- $(x +y)+ xx(x +y) y log *. 
Prob. 102. Solve (D 1 + $DD' + 6D n )z = (y - 2 X y\ 

Ans. z = (f>(y 2 x) + ip(y 3^) + x log (y 2x). 

tfz &z $z 
Prob. 103. Solve ^-; ^-^- -f ^ -- 2 = 0. 



Prob. 104. Show that for an equation of the form (15) the solu- 
tion given by equation (19) is equivalent to equation (16). 

i tfz i fiz i d*z i dz . 
Prob. 105. Solve ^-5 -- 1^- = ^1 -- ;^~ by transpose 

x dx x dx y dy y 9 dy 

tion to the independent variables x* an 
Prob.,06. So,ve,'l 



INDEX. 



Auxiliary system of lines, 51, 55. 

Besselian functions, 44 note. 
Bessel's equation, 43. 
Boundary (of real solutions), 17. 

Characteristics, 59. 
Clairaut's equation, 22, 61. 
Complementary function, 35, 48. 
Complete integral, 2, 30, 57, 60. 
Condition of integrability, 50. 
Cusp-locus, 17, 20. 

Decomposable equations, 13. 
Differentiation, solution by, 20. 
Direct integration, 2. 
Discriminant, 16, 18. 
Doubly infinite systems, 26, 31. 

Envelope, 15. 

Equation of energy, 29, 49. 

Equipotential surfaces, 52 note. 

Exact differentials, 2, 51. 

Exact equations, 6, 27, 30, 36. 

Extension of the linear equation, 12. 

Finite solutions, 45 

First integral, 30, 31. 

First order and degree, i, 50. 

First order and second degree, 12. 

General integral, 53, 57. 
Geometrical applications, 23, 31. 
Geometrical representation of a differ 
ential equation, 3, 13, 15, 26. 

Homogeneous equations, 9. 
Homogeneous linear equations, 40 

Integral, 26, 34, 49, 66. 
Integral equation, 26. 
Integral surface, 58. 
Integrating factors, 8, n, 33. 

Jacobians, 55 note. 



Lagrange's lines, 55. 

Lagrange's solution, 53, 56. 

Linear elements of surface, 59 note. 

Linear equations, 10, 34, 36. 

Linear partial differential equation, 66. 

Logarithmic solutions, 44 note, 46 note. 

Monge's method, 65, Prob. 97 note. 
Monge's solution, 53, Prob. 79 

Node-locus, 19. 
Non-integrable equation, 51. 

Operative symbols, 36, 41. 
Order, equations of first, i. 

of second, 28. 

Orthogonal surfaces, 52 note, 
Orthogonal trajectories, 24. 

Parameters or arbitrary constants, 4, 

15, 26, 31, 60. 
Partial differential equations. 

first order and degree, 53. 

linear, 66. / 

second order, 63 
Particular integral. 2 

determined in series, 46. 

of linear equation, 35, 38. 41 
Pencil of curves, 14. 
Primitive 5 55 

Separation of variables. 2, 51. 
Series, solutions in, 42, 
Simultaneous equations, 25, 47 
Singular solutions, 15, 18, 26 note, 59. 
Symbolic solutions, 37 et seq., 41, 67. 
Systems of curves, 4, 26, 31. 
Systems of differential equations, 47. 

Tac locus, 1 6 

Trajectories, 23 

Transcendental and algebraic forms of 

solution, 2. 
Transformation of linear equations, 46. 

Ultimate intersections, 19, 59. 



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